Improved Bonferroni Inequalities via Abstract Tubes: Inequalities and Identities of Inclusion-Exclusion Type

Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1826 3 Berlin Hei...

Author: Klaus Dohmen

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Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1826

3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo

Klaus Dohmen

Improved Bonferroni Inequalities via Abstract Tubes Inequalities and Identities of Inclusion-Exclusion Type

13

Author Klaus Dohmen Department of Mathematics Mittweida University of Applied Sciences Technikumplatz 17 09648 Mittweida Germany e-mail: [email protected] http://www.dohmen.htwm.de

Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

Mathematics Subject Classiﬁcation (2000): primary: 05A19, 05A20, 60C05, 60E15 secondary: 05A15, 05B35, 05C15, 06A15, 62N05, 68M15, 68R10, 90B15, 90B25 ISSN 0075-8434 ISBN 3-540-20025-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10957806

41/3142/du - 543210 - Printed on acid-free paper

Preface This work is based on my habilitation thesis which I prepared at Berlin’s Humboldt-University while I was an assistant professor at the computer science department in the years 1994–2000. The material has been re-arranged, some parts have been signiﬁcantly shortened, while other parts have been expanded. I am especially grateful to Professor Egmar R¨odel at Humboldt-University for giving me the opportunity to work on this subject. Special thanks go to the referees of my habilitation thesis as well as to the anonymous referees of Springer-Verlag for their worthy suggestions that resulted in an improvement of the manuscript. Particular thanks are owed to Professor Douglas Shier (Clemson University) for his valuable comments. Finally, I would like to thank my wife Imke and my children Sophie, Emily and Justus for their never-ending patience. Mittweida, Germany July, 2003

Klaus Dohmen

Contents

1 Introduction and Overview

1

2 Preliminaries 2.1 Graphs and posets . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 7

3 Bonferroni Inequalities via Abstract Tubes 3.1 An outline of abstract tube theory . . . . . . . . . . . 3.1.1 The notion of an abstract tube . . . . . . . . . 3.1.2 Two fundamental theorems on abstract tubes . 3.1.3 An importance sampling scheme . . . . . . . . 3.2 Examples of abstract tubes . . . . . . . . . . . . . . . 3.2.1 Abstract tubes for convex polyhedra . . . . . . 3.2.2 Abstract tubes for unions of balls and spherical

. . . . . . . . . . . . . . . . . . caps

. . . . . . .

. . . . . . .

. . . . . . .

9 9 9 11 13 14 14 15

4 Abstract Tubes via Closure and Kernel Operators 4.1 Abstract tubes via closure operators . . . . . . . . . 4.1.1 Closure operators . . . . . . . . . . . . . . . . 4.1.2 How closure operators lead to abstract tubes 4.2 Abstract tubes via kernel operators . . . . . . . . . . 4.2.1 Kernel operators . . . . . . . . . . . . . . . . 4.2.2 How kernel operators lead to abstract tubes . 4.3 Alternative proofs . . . . . . . . . . . . . . . . . . . 4.3.1 Improved identities via closure operators . . . 4.3.2 Improved inequalities via kernel operators . . 4.3.3 Additional proofs of identities . . . . . . . . . 4.4 The chordal graph sieve . . . . . . . . . . . . . . . .

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. . . . . . . . . . .

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19 19 19 21 25 25 26 28 28 33 35 37

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

5 Recursive Schemes 44 5.1 Recursive schemes for semilattices . . . . . . . . . . . . . . . . . 44 5.2 Dynamic programming approach . . . . . . . . . . . . . . . . . . 45

VIII

Contents

6 Reliability Applications 6.1 System reliability . . . . . . . . . . . . . . . . . . . 6.1.1 Terminology . . . . . . . . . . . . . . . . . 6.1.2 Abstract tubes for system reliability . . . . 6.1.3 Shier’s pseudopolynomial algorithm . . . . 6.1.4 Inclusion-exclusion and domination theory . 6.2 Special reliability systems . . . . . . . . . . . . . . 6.2.1 Communications networks . . . . . . . . . . 6.2.2 k-out-of-n systems . . . . . . . . . . . . . . 6.2.3 Consecutive k-out-of-n systems . . . . . . . 6.3 Reliability covering problems . . . . . . . . . . . . 6.3.1 The hypergraph model . . . . . . . . . . . . 6.3.2 Abstract tubes and polynomial algorithms . 6.4 Chapter notes . . . . . . . . . . . . . . . . . . . . . 7 Combinatorial Applications and Related Topics 7.1 Inclusion-exclusion on partition lattices . . . . . 7.2 Chromatic polynomials and broken circuits . . . 7.2.1 The usual chromatic polynomial . . . . . 7.2.2 The generalized chromatic polynomial . . 7.3 Sums over partially ordered sets . . . . . . . . . . 7.3.1 A general theorem on sums . . . . . . . . 7.3.2 Application to inclusion-exclusion . . . . 7.4 Matroid polynomials and the β invariant . . . . . 7.4.1 The Tutte polynomial . . . . . . . . . . . 7.4.2 The characteristic polynomial . . . . . . . 7.4.3 The β invariant . . . . . . . . . . . . . . . 7.5 Euler characteristics and M¨obius functions . . . . 7.5.1 Euler characteristics . . . . . . . . . . . . 7.5.2 M¨ obius functions . . . . . . . . . . . . . .

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47 47 47 48 49 52 53 53 66 70 75 75 75 81

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82 82 83 84 86 89 89 90 91 91 93 94 95 95 96

Bibliography

100

Author Index

111

Subject Index

113

1

Introduction and Overview

Many problems in combinatorics, number theory, probability theory, reliability theory and statistics can be solved by applying a unifying method, which is known as the principle of inclusion-exclusion. The principle of inclusionexclusion expresses the indicator function of a union of ﬁnitely many sets as an alternating sum of indicator functions of their intersections. More precisely, for any ﬁnite family of sets {Av }v∈V the principle of inclusion-exclusion states that (1.1)

χ

v∈V

Av

=

I⊆V I=∅

|I|−1

(−1)

χ

Ai

,

i∈I

where χ(A) denotes the indicator function of A with respect to some universal set Ω, that is, χ(A)(ω) = 1 if ω ∈ A, and = 0 if ω ∈Ω\A. Equivalently, χ(A)(ω) (1.1) can be expressed as χ v∈V Av = I⊆V (−1)|I| χ i∈I Ai , where Av denotes the complement of Av in Ω and i∈∅ Ai = Ω. A proof by induction on the number of sets is a common task in undergraduate courses. Usually, the Av ’s are measurable with respect to some ﬁnite measure µ on a σ-ﬁeld of subsets of Ω. Integration of the indicator function identity (1.1) with respect to µ then gives |I|−1 µ (1.2) Av = (−1) µ Ai , v∈V

I⊆V I=∅

i∈I

which now expresses the measure of a union of ﬁnitely many sets as an alternating sum of measures of their intersections. The step leading from (1.1) to (1.2) is referred to as the method of indicators [GS96b]. Naturally, two special cases are of particular interest, namely the case where µ is the counting measure on the power set of Ω, and the case where µ is a probability measure on a σ-ﬁeld of subsets of Ω. These special cases are sometimes attributed to Sylvester (1883) and Poincar´e (1896), although the second edition of Montmort’s book “Essai d’Analyse sur les Jeux de Hazard”, which appeared in 1714, already contains an implicit use of the method, based on a letter by N. Bernoulli in 1710. A ﬁrst K. Dohmen: LNM 1826, pp. 1–4, 2003. c Springer-Verlag Berlin Heidelberg 2003

2

1 Introduction and Overview

explicit description of the method was given by Da Silva (1854). For references to these sources and additional historical notes, we refer to Tak´ acs [Tak67]. Since the identities (1.1) and (1.2) contain 2|V | − 1 terms and intersections of up to |V | sets, one often resorts to inequalities like that of Boole [Boo54]: χ (1.3) Av ≤ χ(Ai ) . v∈V

i∈V

A more general result, ﬁrst discovered by Ch. Jordan [Jor27] and later by Bonferroni [Bon36], states that for any ﬁnite family of sets {Av }v∈V and any r ∈ , (1.4) (r odd), Av ≤ (−1)|I|−1 χ Ai χ

v∈V

χ

(1.5)

I⊆V 0<|I|≤r

Av

≥

(−1)|I|−1 χ

I⊆V 0<|I|≤r

v∈V

i∈I

Ai

(r even).

i∈I

Nowadays, these inequalities are usually referred to as Bonferroni inequalities. Again, there is no real restriction in using indicator functions rather than measures, since these inequalities can be integrated with respect to any ﬁnite measure µ (e.g., a probability measure) on any σ-ﬁeld containing the sets Av , v ∈ V . Numerous extensions of the classical Bonferroni inequalities (1.4) and (1.5) were established in the second half of the 20th century. An excellent survey of the various results, applications and methods of proof is given in the recent book of Galambos and Simonelli [GS96b]. The following inequalities due to Galambos [Gal75], which are valid for any ﬁnite collection of sets {Av }v∈V , improve (1.4) and (1.5) by including additional terms based on the (r + 1)-subsets of V : χ χ

v∈V

≤

Av

v∈V

|I|−1

(−1)

χ

I⊆V 0<|I|≤r

Av

≥

I⊆V 0<|I|≤r

|I|−1

(−1)

χ

Ai

i∈I

i∈I

Ai

r+1 − χ |V | I⊆V |I|=r+1

r+1 + χ |V | I⊆V |I|=r+1

Ai

i∈I

(r odd),

Ai

(r even).

i∈I

Inequalities for the measure or indicator function of a union which are valid for any ﬁnite collection of sets like the preceding ones are frequently referred to as Bonferroni-type inequalities [GS96a, GS96b] or as inequalities of BonferroniGalambos type [MS85, M˘ ar89, TX89]. A new inequality of Bonferroni-Galambos type based on chordal graphs will be established in Chapter 4 of this book. The main part of this work deals with improved Bonferroni inequalities and improved inclusion-exclusion identities that require the collection of sets to satisfy some structural restrictions. Examples of such well-structured collections of sets arise in some problems of statistical inference [NW92, NW97], reliability

1 Introduction and Overview

A2

3

A4

A1

A5

A3

Figure 1.1: A Venn diagram of ﬁve sets. theory [Doh98, Doh03, GNW02] and chromatic graph theory [Doh99a, Doh99d]. We shall mainly be interested in inclusion-exclusion identities of the form |I|−1 Av = (−1) χ Ai χ v∈V

i∈I

I∈

and inequalities of type (1.6) χ Av ≤ (−1)|I|−1 χ Ai (1.7)

χ

v∈V

v∈V

I∈ |I|≤r

Av

≥

I∈ |I|≤r

(−1)|I|−1 χ

i∈I

(r odd),

Ai

(r even),

i∈I

where is a restricted set of non-empty subsets of V , and where (1.6) and (1.7) are at least as sharp as their classical counterparts (1.4) and (1.5). A ﬁrst straightforward approach is to account only for non-empty subsets I of V satisfying i∈I Ai = ∅. In fact, Lozinskii [Loz92] demonstrates that a skillful implementation of this approach leads to a reduction of the average running time of the standard inclusion-exclusion algorithm for counting satisfying assignments of propositional formulae in conjunctive normal form. In the present work, however, we are interested in more subtle improvements that arise from logical dependencies of the sets involved. Consider for instance the ﬁve sets A1 –A5 , whose Venn diagram is shown in Figure 1.1. The classical inclusion-exclusion identity for the indicator function of the union of these sets gives a sum of 25 − 1 = 31 terms, many of which are equal with opposite sign. After cancelling out, we are left with χ(A1 ∪ A2 ∪ A3 ∪ A4 ∪ A5 ) = χ(A1 ) + χ(A2 ) + χ(A3 ) + χ(A4 ) + χ(A5 ) − χ(A1 ∩ A2 ) − χ(A2 ∩ A3 ) − χ(A3 ∩ A4 ) − χ(A4 ∩ A5 ) , which contains only 9 terms. Our purpose is to predict such cancellations in advance and thus to obtain shorter inclusion-exclusion identities and sharper Bonferroni inequalities for the indicator function of a union.

4

1 Introduction and Overview

The book is organized as follows: In Chapter 2 we review some basic notions of graph theory, lattice theory and combinatorial topology. Readers with an appropriate background in these disciplines may want to skip this chapter. Then, in Chapter 3, we give an introduction to the upcoming theory of abstract tubes, which was initiated by Naiman and Wynn [NW92, NW97], and which serves as a framework for establishing improved inclusion-exclusion identities as well as improved Bonferroni inequalities which are provably at least as sharp as their classical counterparts. Some appealing geometric examples of abstract tubes demonstrate the applicability of this beautyful new theory. In Chapter 4 the main results of abstract tube theory in Chapter 3 are applied in establishing improved inclusion-exclusion identities and Bonferroni inequalities associated with closure and kernel operators. Several results from the literature such as the semilattice sieve of Narushima [Nar74, Nar82] and the tree sieve of Naiman and Wynn [NW92] are thus rediscovered in a uniﬁed way. We also provide some elementary alternative proofs (not using abstract tubes) as well as a generalization of one of the identities. One of these alternative proofs uses Zeilberger’s abstract lace expansion [Zei97], which turns out to be a valuable tool in establishing new inclusion-exclusion expansions. Inspired by our results on closure operators and abstract tubes, we derive a new generally-valid inequality of Bonferroni-Galambos type, where the selection of intersections in the estimates is determined by a chordal graph. By varying this graph, several well-known and new Bonferroni-type inequalities are obtained in a uniﬁed way. In Chapter 5 we deduce some recursive schemes from our inclusion-exclusion results in Chapter 4, which will then be used in Chapter 6 in deriving Shier’s recursive algorithm and semilattice expression for system reliability [Shi88, Shi91]. In Chapter 6 our results of the preceding chapters are applied to reliability analysis of coherent binary systems such as communications networks, k-out-ofn systems and consecutive k-out-of-n systems. We then turn our attention to reliability covering problems and identify a comprehensive class of hypergraphs for which the coverage probability can be computed in polynomial time. In Chapter 7, which is devoted to applications to combinatorics and related topics, we give an abstract tube generalization of Narushima’s principle of inclusion-exclusion on partition lattices [Nar74, Nar77] and of Whitney’s broken circuit theorem on the chromatic polynomial of a graph. The results on the chromatic polynomial are then extended to a new two-variable polynomial generalizing the chromatic polynomial, the independence polynomial, and the matching polynomial. After generalizing our inclusion-exclusion identity for kernel operators to a result on sums over partially ordered sets, similar results as for the chromatic polynomial are obtained for the Tutte polynomial, the characteristic polynomial and the β invariant of a matroid, the Euler characteristic of an abstract simplicial complex and the M¨obius function of a ﬁnite lattice.

2

Preliminaries

In this chapter we review some notions from discrete mathematics and combinatorial topology. Some particular notions are deﬁned later when they are ﬁrst needed. Readers with an appropriate background may want to skip this chapter.

2.1

Graphs and posets

Unless stated otherwise, all graphs are ﬁnite, undirected and without loops or multiple edges. When directed edges or loops and multiple edges are allowed, we respectively speak of a digraph or multigraph rather than a graph. Definition 2.1.1 A graph is a pair G = (V, E) where V = V (G) is a ﬁnite set, whose elements are the vertices of the graph, and where E = E(G) is a set of two-element subsets of V , whose elements are the edges of the graph. We write n(G) and m(G) to denote the number of vertices and edges of G, respectively. If e = {x, y} is an edge of G, then e is said to join x and y, and x and y are called adjacent in G. G is complete if any two distinct vertices of G are adjacent in G. The neighborhood NG (v) of a vertex v of G is the set of all vertices w of G such that v and w are adjacent in G; the cardinality of NG (v) is called the degree of v. A subgraph of G is a graph G such that V (G ) ⊆ V (G) and E(G ) ⊆ E(G). With any subset I of V (G) we associate the vertex-induced subgraph G[I] := (I, {e ∈ E(G) | e ⊆ I}). Similarly, any subset J of E(G) with

we associate the edge-subgraph G[J] := J, J where J is a shorthand for

j. A subset X of V (G) is independent or stable if G[X] is edgeless. A j∈J clique of G is a subset X of V (G) such that G[X] is complete. The clique number of G is the maximum cardinality of a clique of G. A vertex v ∈ V (G) is referred to as a simplicial vertex of G if its neighborhood is a clique. A path between x and y or x, y-path in a graph G is a sequence (v1 , . . . , vk ) where k ≥ 1 and where v1 , . . . , vk are distinct vertices of G such that x = v1 , y = vk and {vi , vi+1 } ∈ E(G) for i = 1, . . . , k − 1. Two paths (v1 , . . . , vk ) and (w1 , . . . , wl ) are independent if vi = wj for i = 2, . . . , k − 1 and j = 2, . . . , l − 1. A cycle of G is a sequence (v1 , . . . , vk , v1 ) where k ≥ 3 and where v1 , . . . , vk are distinct vertices of G such that {vi , vi+1 } ∈ E(G) for i = 1, . . . , k − 1 and K. Dohmen: LNM 1826, pp. 5–8, 2003. c Springer-Verlag Berlin Heidelberg 2003

6

2 Preliminaries

{vk , v1 } ∈ E(G). The length of a path (v1 , . . . , vk ) resp. cycle (v1 , . . . , vk , v1 ) is k. Throughout, paths and cycles of G are viewed as subgraphs of G. A chord of a path P of G is an edge of G joining two vertices of P that are not adjacent in P ; similarly for cycles. G is chordal or triangulated if any cycle of G of length greater than three has a chord. G is connected if there is a path between any pair of vertices of G. A connected component of G is a maximal connected subgraph of G. The number of connected components of G is denoted by c(G). G is called biconnected if there are at least two independent paths between any pair of vertices of G. A maximal biconnected subgraph of G is referred to as a block of G. G is called a block graph if every block of G is complete. A tree is a connected graph without cycles. Any connected subgraph of a tree is called a subtree. Vertices of degree 0 or 1 in a tree are called leaves. A subgraph G of a graph G is a spanning tree of G if G is a tree and V (G) = V (G ). Two graphs G = (V, E) and G = (V , E ) are isomorphic if there is a bijective mapping φ : V → V such that {v, w} ∈ E(G) iff {φ(v), φ(w)} ∈ E(G ). The join G1 ∗ G2 of two vertex-disjoint graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ) is deﬁned as G1 ∗ G2 := (V1 ∪ V2 , E1 ∪ E2 ∪ {{v1 , v2 } | v1 ∈ V1 , v2 ∈ V2 }). If G1 and G2 are not vertex-disjoint, then in the preceding deﬁnition they are replaced by vertex-disjoint isomorphic copies G1 and G2 . Thus, G1 ∗ G2 is unique up to isomorphism. Finally, for each n ∈ we use Kn , Ln and Pn to denote the complete graph, the edgeless graph and the path on n vertices, respectively. Our terminology on posets agrees with that of Gr¨ atzer [Gra98]. Definition 2.1.2 Let P be a set. A partial ordering relation on P is a binary relation which is reﬂexive, antisymmetric and transitive. Given such a relation, we refer to P as a partially ordered set or poset. If there are no ambiguities we use ≤ to denote the partial ordering relation on P and write a < b if a ≤ b and a = b for any a, b ∈ P . Two elements a, b ∈ P are called comparable if a ≤ b or b ≤ a; otherwise, they are called incomparable. A subset C of P is a chain resp. antichain of P if any two elements of C are comparable resp. incomparable. In the particular case where P itself is a chain the partial ordering relation on P is called linear. The length of a chain C of P is one less than the cardinality of C. For any p ∈ P the length of a longest chain extending upward to p is called the height of p. The height of P is the maximum height of an element of P . P is called lower-finite resp. upper-finite if {x ∈ P | x ≤ p} resp. {x ∈ P | x ≥ p} is ﬁnite for any p ∈ P . A lower bound of a subset X of P is an element p ∈ P such that p ≤ x for any x ∈ X. Dually, an upper bound of a subset X of P is an element p ∈ P such that p ≥ x for any x ∈ X. A lower semilattice is a partially ordered set P such that P contains a greatest lower bound for any non-empty ﬁnite subset X of P , which is called the infimum or meet of X and abbreviated to inf X or X. Dually, an upper semilattice is a partially ordered set P such that P contains a least upper bound for any non-empty ﬁnite subset X of P , which is called the supremum or join of X and abbreviated to sup X or X. The infimum and supremum of any two elements x, y ∈ P are denoted

2.2 Topology

7

by x ∧ y and x ∨ y, respectively. An element p of a lower semilattice P is called meet-irreducible if p = x ∧ y implies p = x or p = y. Dually, an element p of an upper semilattice P is called join-irreducible if p = x ∨ y implies p = x or p = y. A lower resp. upper subsemilattice of a lower resp. upper semilattice P is a subset X of P which is closed under ∧ resp. ∨. The lower resp. upper subsemilattice of P generated by a subset X of P is the intersection of all lower resp. upper subsemilattices of P which include X as a subset. A lattice is a partially ordered set P which is both a lower and an upper semilattice. We write L = [ˆ 0, ˆ 1] to signify that L has a least element ˆ0 and greatest element ˆ1. A bijective mapping φ from a lattice L to a lattice L is a lattice isomorphism if φ(x ∧ y) = φ(x) ∧ φ(y) and φ(x ∨ y) = φ(x) ∨ φ(y) for any x, y ∈ L. Two lattices L and L are isomorphic if there exists a lattice isomorphism φ : L → L . Usually, a partially ordered set P is represented as a Hasse diagram in which elements of P are represented by points in the plane, and points associated with distinct elements x and y of P are joined by a line segment ascending from x to y if x < y and if there is no p ∈ P lying strictly between x and y. Thus, relations implied by transitivity are not explicitly shown. Figure 7.3 on page 99 shows a Hasse diagram for ﬁve elements a, b, c, d, e where d < b, e and b < a, c.

2.2

Topology

In this section, we review the topological concepts needed throughout this text. Readers which are unfamiliar with topology should consult an easily accessable textbook like that of Harzheim [Har78], Rotman [Rot88], or Spanier [Spa66]. Definition 2.2.1 Let V be a ﬁnite set. An abstract simplicial complex on V is any set of non-empty subsets of V such that I ∈ and ∅ = J ⊂ I imply J ∈ . The elements of are called faces or simplices of , while the elements of I∈ I are called vertices of . Throughout this text, we use Vert( ) to denote the set of vertices of . By deﬁnition, the dimension of a face I, dim I, is one less than its cardinality. The dimension of , dim , is the maximum dimension of a face of . The Euler characteristic γ( ) is deﬁned by (2.1)

γ( ) := c0 ( ) − c1 ( ) + c2 ( ) − c3 ( ) + · · ·

where ck ( ) denotes the number of faces of dimension k of (k = 0, 1, 2, 3, . . . ). A geometric realization of is any topological space homeomorphic to

π (2.2) := ti eπi ti ≥ 0 ∀ i ∈ I and ti = 1 , I∈

i∈I

i∈I

where π is an injective mapping from Vert( ) to {1, . . . , n} and where {e1 , . . . , en } is the standard basis of n (here, n is considered as a vector space). By deﬁnition, two topological spaces X and Y are homeomorphic if there is a continuous bijective mapping φ : X → Y whose inverse φ−1 is continuous as well. Thus, a

8

2 Preliminaries

geometric realization of is unique up to homeomorphism. A topological space X is contractible if there is a continuous mapping F : X × [0, 1] → X such that F (x, 0) = x for any x ∈ X and F ( · , 1) ≡ c for some constant c ∈ X. Since any two homeomorphic spaces are either both contractible or not, we may call an abstract simplicial complex contractible if it has a contractible realization. Example 2.2.2 For any non-empty ﬁnite set V the abstract simplicial complex

∗ (V ) consisting of all non-empty subsets of V is contractible. In fact, the geometric realization (2.2) of ∗ (V ) is contractible by means of the mapping F :

π

× [0, 1] →

π

, (x, t) → (1 − t)x + teπv ,

where v ∈ V can be chosen arbitrarily. Example 2.2.3 Figure 2.1 shows a geometric realization of the abstract simplicial complex = {{a}, {b}, {c}, {d}, {e}, {f }, {g}, {a, b}, {a, c}, {b, c}, {a, b, c}, {c, d}, {c, g}, {d, e}, {d, e, f }, {d, f }, {d, g}, {e, f}} . This complex is not contractible because of the unshaded hole in the middle. However, if we ﬁll-in the hole (that is, if we attach the face {c, d, g} to the complex), then a contractible abstract simplicial complex would result. a

e g f c

d

b

Figure 2.1: A geometric realization of an abstract simplicial complex.

3 Bonferroni Inequalities via Abstract Tubes In the 1990s Naiman and Wynn [NW92, NW97] introduced the topological concept of an abstract tube in order to obtain improved inclusion-exclusion identities and Bonferroni inequalities for a ﬁnite family of events. Roughly speaking, an abstract tube is a ﬁnite collection of sets together with an associated abstract simplicial complex. The signiﬁcance of this new concept is due to the fact that any abstract tube gives rise to an improved inclusion-exclusion identity and a series of improved Bonferroni inequalities where the selection of intersections in the estimates is determined by the associated abstract simplicial complex. In this chapter we review the basic notions and facts of abstract tube theory as they appear in Naiman and Wynn [NW97]. We then turn to some appealing geometric examples that demonstrate the applicability of this new theory.

3.1 3.1.1

An outline of abstract tube theory The notion of an abstract tube

The following deﬁnition is due to Naiman and Wynn [NW97]. Here and elsewhere in this text, we use ∗ (V ) to denote the set of non-empty subsets of V . Definition 3.1.1 An abstract tube is a pair (, ) consisting of a ﬁnite family of sets =

{Av }v∈V and an abstract simplicial complex ⊆ ∗ (V ) such that for any ω ∈ v∈V Av the abstract simplicial complex I ∈ ω ∈ Ai

(3.1)

(ω) :=

i∈I

is contractible. An abstract tube (1 , 1 ) is referred to as a subtube of an abstract tube (2 , 2 ) if 1 = 2 and 1 ⊆ 2 . K. Dohmen: LNM 1826, pp. 9–18, 2003. c Springer-Verlag Berlin Heidelberg 2003

10

3

Bonferroni Inequalities via Abstract Tubes

Remark. If we replace “is contractible” by “has Euler characteristic 1”, then we get the weaker deﬁnition used in [NW92]. However, in order to establish improved Bonferroni inequalities the stronger deﬁnition of [NW97] is needed.

Example 3.1.2 Let {Av }v∈V be a ﬁnite family of sets. Then, ({Av }v∈V , ∗ (V )) is an abstract tube where ∗ (V ) again denotes the set of non-empty subsets of V . Example 3.1.3 Consider the sets A1 , A2 , A3 , A4 , A5 , whose Venn diagram is shown in Figure 1.1. It is straightforward to check that {A1 , A2 , A3 , A4 , A5 }, {{1}, {2}, {3}, {4}, {5}, {1, 2}, {2, 3}, {3, 4}, {4, 5}} is an abstract tube. Example 3.1.4 Let {Av }v∈V be a ﬁnite family of sets. Trivially, {Av }v∈V , I ∈ ∗ (V ) Ai = ∅

i∈I

is an abstract tube. The latter example gives rise to the following deﬁnition: Definition 3.1.5 Let = {Av }v∈V be a ﬁnite family of sets. Then the set of all I ∈ ∗ (V ) satisfying i∈I Ai = ∅ is called the nerve of . In view of Example 3.1.4, any ﬁnite family of sets forms an abstract tube with its nerve. In Section 3.2 and Chapter 4 we will discover some more subtle classes of abstract tubes. The following notion of a weak abstract tube, which is equally due to Naiman and Wynn [NW97], slightly generalizes the notion of an abstract tube. Definition 3.1.6 Let (Ω, , µ) be a measure space. Then a pair (, ) consisting of a ﬁnite collection of sets = {Av }v∈V ⊆ and an abstract simplicial complex ⊆ ∗ (V ) is called a weak abstract tube with

respect to µ if (ω), as deﬁned in (3.1), is contractible for almost every ω ∈ v∈V Av with respect to µ. Remark. If (Ω, , P ) is a probability space and (, ) is a weak abstract tube with respect to P , then the mapping ω → (ω) may be considered as a random abstract simplicial complex ran which is required to be P -almost surely contractible.

3.1 An outline of abstract tube theory

3.1.2

11

Two fundamental theorems on abstract tubes

In the following, we restate the main results of abstract tube theory due to Naiman and Wynn [NW97] and essentially give their original proofs. We start with two preliminary propositions, which are of vital importance. The ﬁrst proposition provides a Bonferroni-type generalization of the topological fact that the Euler characteristic of any contractible abstract simplicial complex equals 1. Proposition 3.1.7 [NW97] Let Then, for any r ∈ , r−1 k=0 r−1

be a contractible abstract simplicial complex.

(−1)k ck ( ) ≤

1

(r even),

(−1)k ck ( ) ≥

1

(r odd).

k=0

Proof. Let Ck denote the free abelian group generated by the faces of dimension k of , where by convention, Ck = (0) for k < 0 or k > dim . Thus, the elements of Ck are formal linear combinations of faces of dimension k of with integer coeﬃcients. Obviously, the rank of Ck , that is, the maximum number of linearly independent elements of inﬁnite order in the group, coincides with the number ck = ck ( ) of faces of dimension k of . Now, ﬁx an arbitrary linear ordering of of , and for any face F of dimension k of kthe vertices i deﬁne ∂k F := i=0 (−1) Fi , where Fi is obtained from F by omitting the (i + 1)-st element of F , when the elements of F are ordered according to the ﬁxed linear ordering of the vertices of . Extending ∂k by linearity to all of Ck gives a homomorphism ∂k : Ck → Ck−1 satisfying ∂k ◦ ∂k+1 = 0 for any k. The k-th homology group Hk of is the quotient of the kernel Zk of ∂k by the image Bk of ∂k+1 . Let hk , zk and bk denote the rank of Hk , Zk and Bk , respectively. (In homology theory, hk is termed the k-th Betti number of .) i

∂

k Bk−1 → 0 (k = 0, 1, . . . ) From the short exact sequences 0 → Zk → Ck −→ πk i and 0 → Bk → Zk −→ Hk → 0 (k = 0, 1, . . . ), where πk is the canonical epimorphism from Zk onto Hk , we obtain zk + bk−1 = ck (k = 0, 1, . . . ) resp. bk + hk = zk (k = 0, 1, . . . ). Thus, hk = ck − bk − bk−1 (k = 0, 1, . . . ), whence

(3.2)

r−1

(−1)k hk =

k=0

r−1

(−1)k ck + (−1)r br−1 ,

k=0

which generalizes the usual Euler-Poincar´e formula. Since br−1 ≥ 0, (3.2) gives r−1 k=0 r−1 k=0

(−1)k hk (−1)k hk

≥ ≤

r−1 k=0 r−1 k=0

(−1)k ck

(r even),

(−1)k ck

(r odd).

12

Since

3

Bonferroni Inequalities via Abstract Tubes

is contractible, h0 = 1 and hk = 0 for k > 0. Hence, the result. 2

Proposition 3.1.8 [NW97] Let and be contractible abstract simplicial complexes where is a subcomplex of , that is, ⊆ . Then, for any r ∈ , r−1 k=0 r−1

(−1)k ck ( ) ≤ (−1)k ck ( ) ≥

k=0

r−1 k=0 r−1

(−1)k ck ( )

(r even),

(−1)k ck ( )

(r odd).

k=0

Proof. As in the preceding proof, let Ck , Hk , Zk and Bk be the groups associated with and ck , hk , zk , bk respectively denote their ranks. Similarly, let ck , hk , zk , bk denote the ranks of the groups Ck , Hk , Zk , Bk associated with . Note that ck = ck ( ) and ck = ck ( ). Since and are contractible, h0 = h0 = 1 and hk = hk = 0 for k > 0. Putting this into (3.2) and the primed version of it, we obtain (3.3)

r−1

(−1)k ck =

k=0

r−1

(−1)k ck + (−1)r−1 br−1 − br−1 .

k=0

as a subgroup of Br−1 , we ﬁnd that br−1 ≤ br−1 , which in By viewing Br−1 combination with (3.3) implies the result. 2

Now, the main results of abstract tube theory, which are due to Naiman and Wynn [NW97], are an immediate consequence of the preceding two propositions. Theorem 3.1.9 [NW97] Let ({Av }v∈V , ) be an abstract tube. Then, for r ∈ , (3.4)

χ

(3.5)

χ

v∈V

v∈V

≥

Av

(−1)|I|−1 χ

I∈ |I|≤r

Av

≤

I∈ |I|≤r

(−1)|I|−1 χ

Ai

i∈I

(r even),

Ai

(r odd).

i∈I

Proof. Choose ω ∈ v∈V Av . By the abstract tube property, (ω) is contractible. The theorem now follows by applying Proposition 3.1.7 with (ω) in place of . 2 Theorem 3.1.10 [NW97] Let ({Av }v∈V , ) and ({Av }v∈V , ) be abstract tubes,

3.1 An outline of abstract tube theory

13

where ({Av }v∈V , ) is a subtube of ({Av }v∈V , ). Then, for any r ∈ ,

|I|−1

(−1)

χ

I∈ |I|≤r

≥

Ai

i∈I

(−1)|I|−1 χ

I∈ |I|≤r

|I|−1

(−1)

χ

I∈ |I|≤r

≤

Ai

(−1)|I|−1 χ

I∈ |I|≤r

i∈I

Ai

i∈I

(r even),

Ai

(r odd).

i∈I

Proof. Choose ω ∈ v∈V Av . From the abstract tube property it follows that both (ω) and (ω) are contractible. The theorem now follows by applying Proposition 3.1.8 with (ω) in place of and with (ω) in place of . 2 Remarks. 1. For the abstract tube ({Av }v∈V , ∗ (V )), Theorem 3.1.9 specializes to the classical Bonferroni inequalities. By virtue of Theorem 3.1.10, the bounds provided by Theorem 3.1.9 are at least as sharp as the classical Bonferroni bounds, since any abstract tube ({Av }v∈V , ) is a subtube of ({Av }v∈V , ∗ (V )). 2. Evidently, in the improved Bonferroni inequalities of Theorem 3.1.9 the equals sign holds if r ≥ dim + 1. In particular, any abstract tube ({Av }v∈V , ) gives rise to a shortened inclusion-exclusion formula where the most complicated intersection involves dim + 1 sets. Due to Naiman and Wynn [NW97], this latter quantity is referred to as the depth of the abstract tube ({Av }v∈V , ). 3. A generalization of Theorem 3.1.9 and Theorem 3.1.10 to weak abstract tubes (see Deﬁnition 3.1.6) is straightforward. In this more general setting, the inequalities hold almost everywhere with respect to some measure.

3.1.3

An importance sampling scheme

By the following theorem the improved Bonferroni inequalities associated with an abstract tube (or weak abstract tube) give rise to an importance sampling scheme for estimating the probability content of a union of ﬁnitely many events. Theorem 3.1.11 [NW97] Let (Ω, , P ) be a probability space, ({Av }v∈V , ) an abstract tube with {Av }v∈V ⊆ , and r ∈ . Define fr : Ω → + , hr : Ω → by 1 fr := Qr hr := χ

χ

I∈ |I|=r+1

v∈V

Av

Ai

, where Qr :=

i∈I

−

I∈ |I|≤r

|I|−1

(−1)

χ

I∈ |I|=r+1

i∈I

Ai

,

P

i∈I

Ai

,

14

3

Bonferroni Inequalities via Abstract Tubes

and let Y1 , Y2 , . . . be independent copies of an fr dP -distributed variable Y . Then, k 1 hr (Yi ) |I|−1 (−1) P Ai + k i=1 fr (Yi ) I∈ i∈I

|I|≤r

is an unbiased estimator for P

v∈V

Av .

Proof. By applying Theorem 3.1.9 ﬁrst for r and then for r + 1 we obtain    r |I|−1 Av − (−1) χ Ai  ≤ χ Ai . 0 ≤ (−1) χ v∈V

I∈ |I|≤r

I∈ |I|=r+1

i∈I

i∈I

Since this is equivalent to 0 ≤ (−1)r hr ≤ fr Qr , we ﬁnd that hr hr (Y ) hr dP = , hr dP = fr dP = Efr fr (Y ) fr >0 fr >0 fr which completes the proof. 2

3.2

Examples of abstract tubes

The original motivation of Naiman and Wynn [NW92, NW97] that led to the theory of abstract tubes was the problem of computing or bounding the volume or probability content of certain geometric objects such as convex polyhedra and unions of ﬁnitely many Euclidean balls or spherical caps, which has applications to the statistical theory of multiple comparisons [NW92] and to computational biology where protein molecules are modeled as unions of balls in three-dimensional Euclidean space [Ede95]. Related problems have also been considered by Edelsbrunner [Ede95] as well as Edelsbrunner and Ramos [ER97].

3.2.1

Abstract tubes for convex polyhedra

In this section we deal with d-dimensional polyhedra in

d .

Definition 3.2.1 A polyhedron in d is a set P = v∈V Hv where {Hv }v∈V is a ﬁnite family of open half-spaces in d and Hv denotes the complement of Hv in d . P is d-dimensional if it contains d + 1 aﬃnely independent points.

Note that due to a law of De Morgan, χ(P ) = 1 − χ v∈V Hv . Thus, by Theorem 3.1.9, the following result of Naiman and Wynn [NW97] gives rise to improved inclusion-exclusion identities and Bonferroni inequalities for the indicator function (and hence for the volume or probability content) of a ddimensional polyhedron in d . For any H ⊆ d we use ∂H to denote the topological boundary of H, that is, the diﬀerence between its topological closure and interior.

3.2 Examples of abstract tubes

15

Theorem 3.2.2 [NW97] Let P = v∈V Hv be a d-dimensional polyhedron in d . Then, {Hv }v∈V , {I ∈ ∗ (V )| i∈I (P ∩ ∂Hi ) = ∅} is an abstract tube.

Proof. (Sketch) Let := I ∈ ∗ (V ) i∈I (P ∩ ∂Hi ) = ∅ and ω ∈ v∈V Hv . By Deﬁnition 3.1.1 we have to prove that (ω), with Hi instead of Ai , is contractible. Observe that (ω) is the nerve of C(ω) := {P ∩ ∂Hi | ω ∈ Hi }, whence by a classical theorem of Borsuk [Bor48] the contractibility of (ω) follows from

that of C(ω). To

prove contractibility of C(ω), we show that some homeomorphic image of C(ω) is contractible: Let π be the homeomorphism mapping

each ν ∈ C(ω) to the intersection of the line segment νω with some hyperplane K separating ω from P . Then, the image of C(ω) under π is convex and hence contractible. 2 Remark. Note that if P is in general position, that is, if no d + 1 facets P ∩ ∂Hi share a common point, then the abstract simplicial complex in the abstract tube of Theorem 3.2.2 is at most (d − 1)-dimensional. In this case, the improved inclusion-exclusion identity associated with the abstract tube of Theorem 3.2.2 involves intersections of up to d half-spaces only and thus contains at most |V | |V | |V | + + ··· + = O |V |d 1 2 d terms, whereas the classical inclusion-exclusion identity still contains 2|V | − 1 terms and involves intersections of up to |V | half-spaces. If P is not in general position, then due to Naiman and ˜Wynn [NW97] it can be perturbed slightly to give a polyhedron P˜ = v∈V H , which is in general position and which v ˜ i ) = ∅} gives rise to a weak abstract tube {Hv }v∈V , {I ∈ ∗ (V )| i∈I (P˜ ∩ ∂ H (involving the original half-spaces Hv ) with respect to Lebesgue measure. Example 3.2.3 Consider the two-dimensional polyhedron P = H1 ∩ H2 ∩ H3 ∩ H4 ∩ H5 which is displayed in Figure 3.1. By combining Theorem 3.2.2 with Theorem 3.1.9 we obtain the improved inclusion-exclusion identity χ(P ) = 1 − χ(H1 ) − χ(H2 ) − χ(H3 ) − χ(H4 ) − χ(H5 ) + χ(H1 ∩ H2 ) + χ(H2 ∩ H3 ) + χ(H3 ∩ H4 ) + χ(H4 ∩ H5 ) , which contains ten terms and intersection of up to two sets only. In contrast, the traditional inclusion-exclusion formula for the indicator function of the same polyhedron contains 25 = 32 terms and intersection of up to ﬁve sets.

3.2.2

Abstract tubes for unions of balls and spherical caps

The abstract tube of Theorem 3.2.5 below derives from a Voronoi subdivision of the underlying space and its associated Delauney complex. We brieﬂy review these notions in the following deﬁnition.

16

3

Bonferroni Inequalities via Abstract Tubes

H5

H4

P

H3 H2 H1

Figure 3.1: A two-dimensional polyhedron.

Definition 3.2.4 Let d ∈ . For any x ∈ d and r > 0 let Bd (x, r) denote the open ball in d with center x and radius r, that is, Bd (x, r) := {y ∈ d | δ(x, y) < r} where δ(·, ·) denotes Euclidean distance. With any ﬁnite set V ⊆ d we associate the Voronoi subdivision of d into non-empty closed convex polyhedra (3.6)

Dv := {x ∈ d | δ(x, v) = min δ(x, u)} u∈V

(v ∈ V )

consisting of points closest in Euclidean distance to v, and the Delauney complex ∗ (V ) := I ∈ (V ) Di = ∅ . (3.7) i∈I

Theorem 3.2.5 [NW92, NW97] Let V be a finite set of points in d and r > 0. Then, ({Bd (v, r)}v∈V , (V )) is an abstract tube.

Proof. (Sketch) Let := (V ) and ω ∈ v∈V Bd (v, r). By Deﬁnition 3.1.1 we have to show that (ω), with Bd (i, r) in place of Ai , is contractible. We ﬁrst observe that (ω) is the nerve of C(ω) := {Di | ω ∈ Bd (i, r)}, whence similar to the proof of

Theorem 3.2.2 the contractibility of (ω) follows from that of C(ω). Indeed, C(ω) is contractible since it is star-shaped with respect to ω. 2 Remark. If the centers of the balls are in general position, that is, if no d + 1 centers of the balls lie in a (d − 1)-dimensional aﬃne subspace of d and no

3.2 Examples of abstract tubes

17

x ∈ d is equidistant from more than d + 1 of the centers of the balls, then the intersection of more than d + 1 of the sets Dv is empty and hence the dimension of (V ) is at most d, regardless of the radius r. In this case, the improved inclusion-exclusion identity associated with the abstract tube of Theorem 3.2.5 involves intersections of up to d + 1 balls and therefore contains at most |V | |V | |V | + + ···+ = O |V |d+1 1 2 d+1 terms, whereas the classical inclusion-exclusion identity still contains 2|V | − 1 terms and involves intersections of up to |V | balls. As noted in [NW92, NW97], the result can be generalized so that the balls may have diﬀerent radii. The following example explains the word tube in Deﬁnition 3.1.1. Example 3.2.6 Figure 3.2 shows ten disks in 2 of equal radius with equidistant centers on a straight line. By Theorem 3.2.5, these ten disks together with the path comprising the centers of the disks constitute an abstract tube.

Figure 3.2: Ten disks of equal radius with centers on a straight line.

Example 3.2.7 Consider the disks in Figure 3.3(a). Clearly, the classical inclusion-exclusion identity for the indicator function of their union contains 220 − 1 = 1048575 terms and intersections of up to 20 disks. In order to apply Theorem 3.2.5 we ﬁrst form the Voronoi subdivision according to (3.6) and then the Delauney complex according to (3.7), see Figure 3.3(b). The Delauney complex contains 20 vertices, 47 edges and 28 triangles. Hence, the improved inclusion-exclusion identity associated with the abstract tube of Theorem 3.2.5 contains only 20 + 47 + 28 = 95 terms and intersections of up to three disks. The preceding theorem has an analogue for the spherical case as described below. This spherical analogue is used in [NW92] to give a simulation method for ﬁnding critical probabilities for multiple-comparisons procedures. Definition 3.2.8 For any x ∈ d, where d denotes the unit d-sphere in d+1 , and any r > 0 we use Bd∗ (x, r) to denote the spherical cap in d with center x and radius r ∈ (0, π/2), that is, Bd∗ (x, r) := {y ∈ d | δ ∗ (x, y) ≤ r}

18

3

Bonferroni Inequalities via Abstract Tubes

(a) 20 disks of equal radius.

(b) Voronoi subdivision and Delauney complex.

Figure 3.3: 20 disks of equal radius and their Delauney complex. where δ ∗ (x, y) := cos−1 x, y is the angular distance between x and y. With any ﬁnite V ⊆ d we associate the spherical Voronoi subdivision of d into regions Dv∗ := {x ∈ d | δ ∗ (x, v) = min δ ∗ (x, u)} u∈V

(v ∈ V )

and the spherical Delauney complex ∗ ∗ (V ) := I ∈ (V ) Di∗ = ∅ . i∈I

Theorem 3.2.9 [NW92, NW97] Let V be a finite set of points in d and r ∈ (0, π/2). If v∈V Bd∗ (v, r) = ∅, then ({Bd∗ (v, r)}v∈V , ∗ (V )) is an abstract tube.

Proof. (Sketch) Let := ∗ (V ) and ω ∈ v∈V Bd∗ (v, r). We have to show that (ω), with Bd∗ (i, r) in place of Ai , is contractible. Similar to the preceding proof, (ω) is the nerve of C ∗ (ω)

:= {Di∗ | ω ∈ Bd∗ (i, r)}, whence the contractibil

ity of (ω) follows from that of C ∗ (ω). In fact, it can be shown that C ∗ (ω) contains every geodesic arc between any of its points and ω, whence it is contractible or equal to the sphere. The second alternative does not apply, since by the requirements the intersection of the spherical caps is empty. 2 Remark. Similar remarks as for the Euclidean case apply to the spherical case. In particular, if the centers of the spherical caps are in general position, that is, if no d + 1 centers lie in a d-dimensional subspace and no x ∈ d is equidistant from more than d + 1 of the centers, then the dimension of ∗ (V ) is at most d.

4 Abstract Tubes via Closure and Kernel Operators In this chapter, the fundamental theorems of abstract tube theory are applied in establishing improved inclusion-exclusion identities and Bonferroni inequalities. We do not state these improved identities and inequalities explicitly, since they can easily be read from Theorem 3.1.9, once an abstract tube is speciﬁed. The key concepts in this chapter are the concept of a closure operator as well as the concept of a kernel operator. In the ﬁrst two sections we review the basic facts on closure operators and kernel operators, and show how these concepts give rise to new abstract tubes and thus to new improvements of the classical inclusion-exclusion principle and its associated truncation inequalities. In the third section we present some elementary alternative proofs (not using abstract tubes) and a generalization of one of the identities. One of the alternative proofs is based on Zeilberger’s abstract lace expansion [Zei97], which turns out to be a valuable tool in deriving new inclusion-exclusion identities. The last section of this chapter is devoted to a Bonferroni-type inequality inspired by one of our results on closure operators and abstract tubes. The main result of this section—the chordal graph sieve—associates a Bonferronitype inequality with any chordal graph G. By varying G, several well-known and new Bonferroni-type inequalities are obtained in a concise and uniﬁed way.

4.1 4.1.1

Abstract tubes via closure operators Closure operators

In what follows it is intuitive to imagine that c is the convex hull operator in d . Here and elsewhere in this text, (V ) is used to denote the power set of V . Definition 4.1.1 Let V be a set. A mapping c : (V ) → (V ) is called a K. Dohmen: LNM 1826, pp. 19–43, 2003. c Springer-Verlag Berlin Heidelberg 2003

20

4

Abstract Tubes via Closure and Kernel Operators

closure operator on V if for all subsets X and Y of V , (i) (ii) (iii)

X ⊆ c(X) (extensionality), X ⊆ Y ⇒ c(X) ⊆ c(Y ) (monotonicity), c(c(X)) = c(X) (idempotence).

A subset X of V is called c-closed if c(X) = X, and c-free if all subsets of X are c-closed. A minimal subset B of X such that c(B) = X is referred to as a c-basis of X. If there are no ambiguities, we supress c in these notions. The following proposition characterizes the free sets by means of their bases. Proposition 4.1.2 [Doh00b] Let c be a closure operator on some finite set V . Then, any subset J of V is free if and only if it is a basis of itself. Proof. We show that any set which is not free is not a basis of itself (the opposite direction is trivial). So assume that J is not free, that is, K ⊂ J for some non-closed set K. If J is not closed, then obviously it can’t be a basis of itself. Thus, we may assume that J is closed. For each k ∈ c(K) \ K we ﬁnd that k ∈ c(K) = c(K \ {k}) ⊆ c(J) = J and hence, J ⊆ c(J \ {k} ∪ {k}) ⊆ c(c(J \ {k}) ∪ {k}) = c(c(J \ {k})) = c(J \ {k}) ⊆ c(J) = J. Therefore, k ∈ J and c(J \ {k}) = J, showing that J is not a basis of itself. 2 The following deﬁnition is due to Edelman and Jamison [EJ85]. Definition 4.1.3 A convex geometry is a pair (V, c) consisting of a ﬁnite set V and a closure operator c on V such that c(∅) = ∅ and such that any c-closed subset of V has a unique c-basis. Convex geometries are sometimes referred to as dual antimatroids (see e.g., [BZ92]). The most prominent example of a convex geometry is the following: Example 4.1.4 [EJ85] Let V be a ﬁnite subset of d , and for any subset X = {x1 , . . . , xn } of V deﬁne c(X) := conv(X) ∩ V where

n n λi xi λ1 , . . . , λn ≥ 0 and λi = 1 . (4.1) conv(X) := i=1

i=1

is the convex hull of X. By the Minkowski-Krein-Milman theorem the extremal points of conv(X) form a unique c-basis for any c-closed subset X of V . Thus, (V, c) is a convex geometry. Some further examples associated with graphs and semilattices follow. Example 4.1.5 [EJ85] For any subset X of some ﬁnite upper semilattice V let c(X) be the smallest upper subsemilattice of V containing X. Then, X is c-closed if and only if X is an upper subsemilattice, and X is c-free if and only

4.1 Abstract tubes via closure operators

21

if X is a chain. Since the set of join-irreducibles of X forms a unique c-basis of any c-closed subset X of V , we conclude that (V, c) is a convex geometry. In a similar way, any lower semilattice gives rise to a convex geometry. Example 4.1.6 [EJ85] For any tree G = (V, E) and any X ⊆ V let c(X) be the set of all z ∈ V lying on the path between x and y for some x, y ∈ X. Then, X is c-closed if and only if G[X] is a tree, and c-free if and only if X = {v, w} for some {v, w} ∈ E or X = {v} for some v ∈ V . Since the leaves of G[X] form a unique c-basis of any c-closed subset X of V , (V, c) is a convex geometry. Example 4.1.7 [EJ85] For any subset X of a connected block graph G = (V, E) let c(X) be the minimal superset Y ⊇ X such that G[Y ] is connected. Then, X is c-closed if and only if G[X] is connected, and c-free if and only if X is a clique. Since the simplicial vertices of G[X] constitute a unique c-basis of any c-closed subset X of V , we conclude that (V, c) is a convex geometry.

4.1.2

How closure operators lead to abstract tubes

The main result of this section is based on the following statement, which can be found as Exercise 8.23c in the textbook of Bj¨ orner and Ziegler [BZ92]. Proposition 4.1.8 [BZ92] Let (V, c) be a convex geometry. Then the abstract simplicial complex consisting of all non-empty c-free subsets of V is contractible. Proof. For a rigorous proof we refer to Edelman and Reiner [ER00]. 2 We are now ready to state the main result of this section. Theorem 4.1.9 [Doh03] Let (V, c) be a convex geometry, and let {Av }v∈V be a finite family of sets such that for any non-empty and non-closed subset X of V , (4.2) Ax ⊆ Av . x∈X

v ∈X /

Then, {Av }v∈V , {I ∈ ∗ (V ) | I c-free} is an abstract tube.

Proof. For ω ∈ v∈V Av let Vω := {v ∈ V | ω ∈ Av } , and for any I ⊆ V deﬁne cω (I) := c(I). Then, by (4.2), Vω is c-closed, whence (Vω , cω ) is a convex geometry. The contractibility of {I ∈ ∗ (V ) | I c-free}(ω) follows from Proposition 4.1.8 and {I ∈ ∗ (V ) | I c-free}(ω) = {I ∈ ∗ (Vω ) | I cω -free}. 2 Remarks. 1. The requirements of Theorem 4.1.9 are already satisﬁed if x∈X Ax ⊆ Av for any non-empty subset X of V and any v ∈ c(X). 2. For c : X → X the abstract tube of Theorem 4.1.9 equals ({Av }v∈V , ∗ (V )). In this case, the associated inequalities are the classical Bonferroni inequalities.

22

4

Abstract Tubes via Closure and Kernel Operators

3. The improved inclusion-exclusion identity associated with the abstract tube of Theorem 4.1.9 involves intersections of at most h(c) := max{|J| : J c-free} sets. A general result of Jamison [JW81] states that h(c) equals the Helly number of the family of c-closed subsets of V . Thus, h(c) is the smallest integer h such that any family of c-closed subsets of V whose intersection is empty has a subfamily of h or fewer sets whose intersection is also empty. 4. Let P be a probability measure on any σ-ﬁeld containing the events Av , v ∈ V , such that for any non-empty and non-closed subset X of V , (4.3) P Ax > 0 and P Av Ax = 1 x∈X

v ∈X /

x∈X

Then, {Av }v∈V , {I ∈ ∗ (V ) | I c-free} is a weak abstract tube in the sense of Deﬁnition 3.1.6. The proof is quite similar and left as an option to the reader. 5. For closure operators c and c on V deﬁne c ≤ c :⇔ c(I) ⊆ c (I) for any subset I of V , or equivalently, c ≤ c :⇔ all c -closed subsets of V are c-closed. Thus, a partial order on the set of closure operators on V is deﬁned. The signiﬁcance of this partial order is as follows. c and c satisfy If c ≤ c and both ∗ the requirements then {Av }v∈V , {I ∈ (V ) | I c -free} is of Theorem 4.1.9, a subtube of {Av }v∈V , {I ∈ ∗ (V ) | I c-free} and hence, by Theorem 3.1.10, the improved Bonferroni inequalities associated with c are at least as sharp as those associated with c. In particular, since the closure operator I → I on V is largest with respect to ≤, the inequalities associated with the abstract tube of Theorem 4.1.9 are at least as sharp as the classical Bonferroni inequalities. We now deduce some results from Theorem 4.1.9. The ﬁrst one is an abstract tube generalization of Narushima’s semilattice sieve [Nar74, Nar77]. Before stating this result, we review the following deﬁnition due to Alexandroﬀ [Ale37]. Definition 4.1.10 The order complex of a ﬁnite partially ordered set V , (V ) for short, is the abstract simplicial complex of all non-empty chains of V . Now the abstract tube generalization of Narushima’s semilattice sieve [Nar74, Nar77] reads as follows. Corollary 4.1.11 [Doh99d] Let {Av }v∈V be a finite family of sets, where V is an upper semilattice such that Ax ∩ Ay ⊆ Ax∨y for any x, y ∈ V . Then, {Av }v∈V and the order complex of V constitute an abstract tube. Proof. Apply Theorem 4.1.9 in connection with Example 4.1.5. 2

4.1 Abstract tubes via closure operators

23

Remark. Corollary 4.1.11 gives the trivial abstract tube if V is a chain. The particular case where Ax ∩ Ay = Ax∨y for any x, y ∈ V is treated in [Doh01a]. The following result is due to Naiman and Wynn [NW92]: Corollary 4.1.12 [NW92] Let {Av }v∈V be a finite family of sets, where the indices form the vertices of a tree G = (V, E) such that Ax ∩ Ay ⊆ Az for any x, y ∈ V and any z on the path between x and y in G. Then, {Av }v∈V and the tree (considered as an abstract simplicial complex) constitute an abstract tube. Proof. Apply Theorem 4.1.9 in connection with Example 4.1.6. 2 Remark. The case where G is a path on n vertices gives rise to the identity n n n χ Ai = χ(Ai ) − χ(Ai−1 ∩ Ai ) , i=1

i=1

i=2

which is valid for all ﬁnite sequences of sets A1 , . . . , An that satisfy Ai ∩Aj ⊆ Ak for i, j = 1, . . . , n and k = i, . . . , j. This latter consequence is again due to Naiman and Wynn [NW92] and explains the cancellations in Figure 1.1. Definition 4.1.13 The clique complex of a graph G is the abstract simplicial complex of all non-empty cliques of G. Since any tree is a connected chordal graph, the following corollary generalizes the preceding one. Corollary 4.1.14 [Doh03] Let {Av }v∈V be a finite family of sets, where the indices form the vertices of a connected chordal graph G = (V, E) such that Ax ∩ Ay ⊆ Az for any x, y ∈ V and any z on any chordless path between x and y. Then, {Av }v∈V and the clique complex of G form an abstract tube. Proof. For any X ⊆ V let c(X) be the set of all z ∈ V lying on a chordless path between x and y for some x, y ∈ X. Then, (V, c) is a convex geometry, where X is free if and only if X is a clique [EJ85, FJ86]. Now apply Theorem 4.1.9. 2 Remarks. 1. Corollary 4.1.14 gives the trivial abstract tube ({Av }v∈V , ∗ (V )) if G is complete, since in this case all subsets of the vertex-set are cliques. 2. Howorka [How81] showed that in chordal graphs where all cycles of length ﬁve have at least three chords the chordless paths are precisely the shortest paths, and that chordal graphs with this property are unions of Ptolemaic graphs and vice versa. Ptolemaic graphs are deﬁned below. For a discussion of this class of graphs and other restricted classes of chordal graphs in connection with convex geometries, the reader is referred to Farber and Jamison [FJ86].

24

4

Abstract Tubes via Closure and Kernel Operators

Definition 4.1.15 A graph G is called Ptolemaic if it is connected and if for any four vertices v, w, x, y of G, d(v, w)d(x, y) ≤ d(v, x)d(w, y) + d(w, x)d(v, y) where d(a, b) denotes the length of a shortest path between a and b in G. Corollary 4.1.16 Let {Av }v∈V be a finite family of sets, where the indices form the vertices of a Ptolemaic graph G = (V, E) such that Ax ∩ Ay ⊆ Az for any x, y ∈ V and any z ∈ V on any shortest path between x and y in G. Then, {Av }v∈V and the clique complex of G constitute an abstract tube. Proof. Corollary 4.1.16 follows from Corollary 4.1.14 and the preceding discussion on Ptolemaic graphs. 2 We proceed with deducing some further consequences of Theorem 4.1.9. Corollary 4.1.17 [Doh03] Let {Av }v∈V be a finite family of sets such that for any non-empty X ⊆ V there is a unique minimal non-empty Y ⊆ X such that (4.4) Ax = Ay . x∈X

y∈Y

Then, {Av }v∈V and the set of all I ∈ ∗ (V ) satisfying i∈I Ai = j∈J Aj for all non-empty proper subsets and supersets J of I constitute an abstract tube. Proof. For any subset X of V deﬁne c(X) = v ∈ V x∈X Ax ⊆ Av if X = ∅, and c(∅) := ∅. Then, c is a closure operator on V . In order to verify that each c-closed subset of V has a unique c-basis, assume that X is non-empty and cclosed. By the above requirement there is a unique minimal non-empty Y ⊆ X satisfying (4.4). By deﬁnition of c, c(X) = c(Y ) ⊆ X and hence, c(Y ) = X. In order to show that Y is smallest with c(Y ) = X, suppose ) = X for that c(Y some other non-empty subset Y of X. By deﬁnition of c, x∈X Ax = y∈Y Ay , whence by the choice of Y , Y ⊇ Y . Consequently, Y is the unique c-basis of X. The corollary now follows from Theorem 4.1.9 and Proposition 4.1.2. 2 Definition 4.1.18 With any ﬁnite set V and any system of non-empty subsets of V we associate the abstract simplicial complex

(V, ) := {I ∈ ∗ (V ) | I ⊇ X for any X ∈ } , consisting of all non-empty subsets of V not including any X ∈ as a subset. Corollary 4.1.19 [Doh99d] Let {Av }v∈V be a finite family of sets where V is linearly ordered. Furthermore, let ⊆ ∗ (V ) such that for any X ∈ , Ax ⊆ Av for some v > max X. (4.5) x∈X

Then, ({Av }v∈V , (V, )) is an abstract tube.

4.2 Abstract tubes via kernel operators

25

Proof. For any X ∈ choose vX such that (4.5) is satisﬁed for v = vX . For any subset I of V deﬁne c1 (I) := c(I) := I ∪ {vX | X ∈ , X ⊆ I}, cn (I) := ∞ cn−1 (c(I)) (n ≥ 2), and c∗ (I) := n=1 cn (I). Then, c∗ is a closure operator on V , and it is easily seen that I \ {vX | X ∈ , X ⊆ I} is the unique c∗ -basis of any c∗ -closed subset I of V . Hence, (V, c∗ ) is a convex geometry. In this convex geometry, any subset I of V is c∗ -free if and only if I ⊇ X for any X ∈ . The result now follows by applying Theorem 4.1.9 with c∗ instead of c. 2 Remark. Note that the corollary gives the trivial abstract tube if is empty, and that the corollary can be dualized by replacing v > max X with v < min X. As a consequence of Corollary 4.1.19 we deduce the following generalization of Corollary 4.1.11. Corollary 4.1.20 [Doh99d] Let {Av }v∈V be a finite family of sets, where V is endowed with a partial ordering relation such that for any x, y ∈ V , Ax ∩Ay ⊆ Az for some upper bound z of x and y. Then, {Av }v∈V and the order complex of V constitute an abstract tube. Proof. The result follows from Corollary 4.1.19 by setting := {{x, y} ⊆ V | x, y incomparable} and considering some arbitrary linear extension. 2 Remark. The inclusion-exclusion identity associated with the abstract tube of Corollary 4.1.20 is due to Narushima [Nar82]. Note, however, that the original requirements of Narushima [Nar82] are slightly stronger in that he requires that for any x, y ∈ V , Ax ∩ Ay ⊆ Az for some minimal upper bound z of x and y.

4.2

Abstract tubes via kernel operators

Similar results as for closure operators are now established for kernel operators. There is, however, no duality between the results of the preceding section and the results of the present section.

4.2.1

Kernel operators

Definition 4.2.1 Let V be a set. A mapping k : (V ) → (V ) is called a kernel operator on V if for all subsets X and Y of V , (i) k(X) ⊆ X

(intensionality),

(ii) X ⊆ Y ⇒ k(X) ⊆ k(Y ) (monotonicity), (iii) k(k(X)) = k(X) (idempotence). A subset X of V is called k-open if k(X) = X.

26

4

Abstract Tubes via Closure and Kernel Operators

There is a well-known correspondence between kernel operators on V and union-closed subsets of the power set of V . Recall that a subset of the power set of V is called union-closed if X ∪ Y ∈ for any X, Y ∈ . For a proof of the following proposition, the reader is referred to the textbook of Ern´e [Ern82]. Proposition 4.2.2 [Ern82] Let V be a finite set. If k is a kernel operator on V , then {X ⊆ V | X k-open} is union-closed. If ⊆ (V ) is union-closed, then I → {X ∈ | X ⊆ I} (I ⊆ V ) defines a kernel operator k on V such that X is k-open if and only if X ∈ .

4.2.2

How kernel operators lead to abstract tubes

Our main result on kernel operators is as follows. Theorem 4.2.3 [Doh00a] Let {Av }v∈V be a finite family of sets, and let k be a kernel operator on V such that for any non-empty and k-open subset X of V , Ax ⊆ Av . x∈X

v ∈X /

Then, {Av }v∈V , {I ∈ ∗ (V ) | k(I) = ∅} is an abstract tube. The proof of Theorem 4.2.3 is based on the following proposition: Proposition 4.2.4 [Doh00a] Let V be a finite set, and let k be a kernel operator on V . Then, the complex {I ∈ ∗ (V ) | k(I) = ∅} is contractible or V is k-open. Proof. Assume that V is not k-open, and let v ∈ V \ k(V ). Then, for any I ⊆ V , v∈ / k (I ∪ {v}), and hence the implication k(I) = ∅ ⇒ k (I ∪ {v}) = ∅ holds for any I ⊆ V . From this we conclude that v is contained in every maximal face of := {I ∈ ∗ (V ) | k(I) = ∅}, whence is a cone and hence contractible. (Recall from topology that a cone is an abstract simplicial complex having a vertex v which is contained in every maximal face of . The geometric realization (2.2) of each such complex is contractible by means of (x, t) → (1 − t)x + teπv .) 2 Proof of Theorem 4.2.3. To obtain a contradiction, assume there is some ω ∈ v∈V Av such that {I ∈ ∗ (V ) | k(I) = ∅}(ω) is not contractible. From {I ∈ ∗ (V ) | k(I) = ∅}(ω) = {I ∈ ∗ (Vω ) | k(I) = ∅} , where Vω := {v ∈ V | ω ∈ Av }, the assumption and Proposition 4.2.4 we conclude that Vω is k-open. On the contrary, the deﬁnition of Vω and the requirements imply that Vω is not k-open. Thus, Theorem 4.2.3 is proved. 2 Remarks. The following remarks relate to Theorem 4.2.3 above.

4.2 Abstract tubes via kernel operators

27

1. By setting k(X) := ∅ for any subset X of V , the abstract tube of Theorem 4.2.3 specializes to the trivial abstract tube. The associated Bonferroni inequalities are the classical Bonferroni inequalities. 2. One can prove that {Av }v∈V , {I ∈ ∗ (V ) | k(I) = ∅} is a weak abstract tube with respect to any probability measure P on the σ-ﬁeld generated by {Av }v∈V such that (4.3) holds for any non-empty and k-open subset X of V . 3. Note that {Av }v∈V , {I ∈ ∗ (V ) | k (I) = ∅} is a subtube of {Av }v∈V , {I ∈ ∗ (V ) | k(I) = ∅} if k and k are as in Theorem 4.2.3 and k ≤ k, where (4.6)

k ≤ k :⇔ k(I) ⊆ k (I) for any subset I of V

or equivalently, (4.7)

k ≤ k :⇔ all k-open subsets of V are k -open.

By this and Theorem 3.1.10, it follows that the improved Bonferroni inequalities associated with k are at least as sharp as those associated with k if k ≤ k. In particular, since the kernel operator I → ∅ on V is largest with respect to ≤, the improved inequalities are at least as sharp as their classical counterparts. The correspondence between kernel operators and union-closed sets leads to the following equivalent formulation of Theorem 4.2.3. Recall from Deﬁnition 4.1.18 that (V, ) := {I ∈ ∗ (V ) | I ⊇ X for any X ∈ }. Theorem 4.2.5 [Doh00a] Let {Av }v∈V a finite family of sets, and let be a union-closed set of non-empty subsets of V such that for any X ∈ , Ax ⊆ Av . x∈X

v ∈X /

Then, ({Av }v∈V , (V, )) is an abstract tube. Proof. The result follows from Proposition 4.2.2 and Theorem 4.2.3. 2 Corollary 4.2.6 [Doh00a] Let {Av }v∈V be a finite family of sets, c a closure operator on V and a set of non-empty subsets of V such that {c(X) | X ∈ } is a chain and such that for any X ∈ , Ax ⊆ Av . (4.8) x∈X

v ∈c(X) /

Then, ({Av }v∈V , (V, )) is an abstract tube.

Proof. Let := { | ∅ = ⊆ }. Then, is union-closed, and for any Y ∈ , Y = {X | X ∈ , X ⊆ Y } ⊆ {c(X) | X ∈ , X ⊆ Y } = c(Z)

28

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Abstract Tubes via Closure and Kernel Operators

for some Z ∈ , Z ⊆ Y . Hence, Z ⊆ Y ⊆ c(Z) and therefore, Ax ⊆ Ax ⊆ Av ⊆ Av . x∈Y

x∈Z

v ∈c(Z) /

v ∈Y /

The result now follows by applying Theorem 4.2.5 with instead of . 2 The following result generalizes Corollary 4.1.19. We refer to [Doh99a] for an exemplary application of the associated inclusion-exclusion identity to counting arrangements with forbidden positions on a chess-like board. Corollary 4.2.7 [Doh99d] Let {Av }v∈V be a finite family of sets, where V is a linearly ordered set, and let ⊆ ∗ (V ) such that for any X ∈ , (4.9) Ax ⊆ Av . x∈X

v>max X

Then, ({Av }v∈V , (V, )) is an abstract tube. Proof. Corollary 4.2.7 follows from Corollary 4.2.6 by using the closure operator X → c(X) where c(X) := {v ∈ V | v ≤ max X} if X = ∅, and c(∅) := ∅. 2

4.3

Alternative proofs

By virtue of Theorem 3.1.9 each abstract tube of the preceding section gives rise to an improved inclusion-exclusion identity and a series of improved Bonferroni inequalities. In this section, we present some elementary alternative proofs for some of these identities and inequalities without making use of abstract tubes.

4.3.1

Improved identities via closure operators

We start with an alternative proof of the inclusion-exclusion identity associated with the abstract tube of Theorem 4.1.9 by using Zeilberger’s abstract lace expansion [Zei97]. We then proceed by generalizing this inclusion-exclusion identity to arbitrary closure operators. Alternative proof by Zeilberger’s abstract lace expansion The following deﬁnition is due to Zeilberger [Zei97]. Definition 4.3.1 [Zei97] Let V be a set. A lace map on V is a mapping l from the power set of V into itself such that for any subsets X and Y of V , (i) l(X) ⊆ X, (ii) l(X) ⊆ Y ⊆ X ⇒ l(X) = l(Y ), (iii)

l(X) = l(Y ) ⇒ l(X ∪ Y ) = l(X).

If l is a lace map on V , then a subset X of V is called an l-lace if l(X) = X.

4.3 Alternative proofs

29

We now state Zeilberger’s abstract lace expansion [Zei97]. Theorem 4.3.2 [Zei97] Let {Av }v∈V be a finite family of sets, and let l be a lace map on V . Then,   χ Av = (−1)|I|−1 χ  Ai ∩ Ak  . (4.10)

I∈ ∗ (V ) I is an l-lace

v∈V

k∈Cl (I)

i∈I

where Cl (I) := {v ∈ V \ I | l(I ∪ {v}) = I} Proof. Applying the inclusion-exclusion principle to the left-hand side we obtain |I|−1 Av = (−1) χ Ai . χ I∈∗ (V )

v∈V

i∈I

From this we conclude that   χ Av = (−1)|J|−1 χ  Aj 

I∈ ∗ (V ) J∈ ∗ (V ) I is an l-lace l(J)=I

v∈V

=

j∈J



(−1)|I|+|J\I|−1 χ 

I∈ ∗ (V ) J∈ ∗ (V ) I is an l-lace l(J)=I

=

I∈ ∗ (V ) I is an l-lace

=

|I|−1

(−1)

χ

I∈ ∗ (V ) I is an l-lace

=

|I|−1

(−1)

I∈ ∗ (V ) I is an l-lace

Ai ∩

i∈I

(−1)|I|−1 χ

χ

Ai

i∈I

i∈I

Ai

i∈I

Ai

j∈J\I

 Aj  

(−1)|J\I|−1 χ 

J∈ ∗ (V ) l(J)=I

|K|−1

χ

(−1)

K⊆V \I l(I∪K)=I

|K|−1

(−1)

χ

K⊆Cl (I)

 Aj 

j∈J\I

Ak

k∈K

Ak

.

k∈K

Now, a second application of the inclusion-exclusion principle gives   χ Av = (−1)|I|−1 χ Ai χ  Ak  , v∈V

I∈ ∗ (V ) I is an l-lace

i∈I

k∈Cl (I)

from which (4.10) immediately follows. 2 In the following, we show that any convex geometry gives rise to a lace map and thus deduce the inclusion-exclusion identity associated with the abstract tube of Theorem 4.1.9 from Zeilberger’s abstract lace expansion.

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Abstract Tubes via Closure and Kernel Operators

Theorem 4.3.3 [Doh02a] Let (V, c) be a convex geometry. For any subset X of V define l(X) := unique c-basis of c(X).

(4.11) Then, l is a lace map.

Proof. We show that l satisﬁes (i)–(iii) in the deﬁnition of a lace map. (i) Clearly, l(X) ⊆ X. (ii) If l(X) ⊆ Y ⊆ X, then c(l(X)) ⊆ c(Y ) ⊆ c(X) by the monotonicity of c. From this and c(l(X)) = c(X) we conclude that c(X) = c(Y ) and hence, l(X) = l(Y ). (iii) From l(X) = l(Y ) we get c(l(X)) = c(l(Y )) and hence, c(X) = c(Y ). Since c is an extensional mapping, X ⊆ c(X) and Y ⊆ c(Y ). Hence, X ∪ Y ⊆ c(X) ∪ c(Y ) = c(X). By this, the monotonicity and idempotence of c we ﬁnd c(X ∪ Y ) = c(X) and hence, l(X ∪ Y ) = l(X). 2 We are now ready to prove the inclusion-exclusion identity |I|−1 (4.12) χ Av = (−1) χ Ai

I∈ ∗ (V ) I c-free

v∈V

i∈I

associated with the abstract tube of Theorem 4.1.9 under the requirements that {Av }v∈V is a ﬁnite family of sets, (V, c) is a convex geometry, and (4.13) Ax ⊆ Av x∈X

v∈c(X)\X

for any non-empty and non-closed subset X of V . Note that (4.13) is more restrictive than the corresponding condition (4.2) of Theorem 4.1.9. Proof of (4.12). For any subset X of V let l(X) be the unique c-basis of c(X) as in (4.11). Then, by Theorem 4.3.3 l is a lace map. We observe that for any subset I ⊆ V , I is an l-lace if and only if I is a c-basis, and that for any l-lace I ⊆ V , Cl (I) = c(I) \ I. Therefore, by applying Theorem 4.3.2 we obtain   χ Av = (−1)|I|−1 χ  Ai ∩ Ak  .

I∈ ∗ (V ) I is a c-basis

v∈V

i∈I

k∈c(I)\I

Now, if I is not a c-basis of itself, then it is not c-closed and hence by (4.13), A ∩ i i∈I k∈c(I)\I Ak = ∅. On the other hand, if I is a c-basis of itself, then it is c-closed and therefore, i∈I Ai ∩ k∈c(I)\I Ak = i∈I Ai . Hence, |I|−1 Av = (−1) χ Ai . χ v∈V

I∈ ∗ (V ) I is a c-basis of itself

i∈I

From this, our claim (4.12) immediately follows since by virtue of Proposition 4.1.2, I is a c-basis of itself if and only if I is c-free. 2

4.3 Alternative proofs

31

Alternative proof for arbitrary closure operators We now establish a generalization of the identity associated with the abstract tube of Theorem 4.1.9 (see (4.12)) for closure operators which not necessarily have the unique basis property. To this end, we need the following deﬁnition. Definition 4.3.4 Let V be a ﬁnite set and c a closure operator on V . A cformation of a subset X of V is any non-empty set of c-bases of X such that

= X. A c-formation of X is odd resp. even if | | is odd resp. even. The c-domination of X, domc (X), is the number of odd c-formations of X minus the number of even c-formations of X. Evidently, if X is not c-closed, then domc (X) = 0, and if X is c-free, then domc (X) = 1. If (V, c) is a convex geometry, then domc (X) = 1 resp. 0 depending on whether X is c-free or not. The following proposition generalizes a result of Edelman and Jamison [EJ85] on convex geometries and a result of Narushima [Nar74, Nar77] on semilattices. Proposition 4.3.5 [Doh99b] Let V be a finite set and c a closure operator on V . Then, for any c-closed subset J of V , (−1)|I| = (−1)|J| domc (J) . I⊆J c(I)=J

Proof. Let J0 , . . . , Jn be the distinct bases of J. Evidently, c(I) = J if and only if Jk ⊆ I ⊆ J for some k ∈ {0, . . . , n}. Thus, by including and excluding terms, (−1)|I| = (−1)||−1 (−1)|I| = (−1)||−1 (−1)|J| . Ë I⊆J ⊆{J0 ,...,Jn } ⊆{J ,...,J n} 0 I : ⊆I⊆J c(I)=J =∅ =∅, Ë=J The result now follows from the deﬁnition of the c-domination domc (J). 2 The following proposition will be used later to derive the main result of this subsection. It may have applications not only to inclusion-exclusion. Proposition 4.3.6 [Doh99b] Let V be a finite set, c a closure operator on V and g a mapping from (V ) into an abelian group such that g = g ◦ c. Then, (−1)|I| g(I) = (−1)|J| domc (J) g(J) . J⊆V J closed

I⊆V

Proof. Since g(I) = g(c(I)) for any subset I of V we ﬁnd that (−1)|I| g(I) = (−1)|I| g(c(I)) = (−1)|I| g(J) . I⊆V

I⊆V

J⊆V I⊆J c(J)=J c(I)=J

By Proposition 4.3.5, the inner sum on the right coincides with (−1)|J| domc (J), which gives the result. 2

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Abstract Tubes via Closure and Kernel Operators

Remarks. In the following remarks, it is assumed that (V, c) is a convex geometry. Thus, domc (X) = 1 if X is c-free, and domc (X) = 0 otherwise. 1. By deﬁning g(I) := 1 for any I ⊆ V where V = ∅, Proposition 4.3.6 gives

(4.14)

(−1)|J| = 0 ,

J⊆V J free

which is an unpublished result due to Lawrence (cf. [EJ85]). , Proposition 4.3.6 shows that 2. By deﬁning g(I) := (−1)|c(I)| for any I ⊆ V in each convex geometry (V, c) there are exactly I⊆V (−1)|c(I)\I| free sets. 3. By deﬁning g(I) := |c(I)| for any I ⊆ V , Proposition 4.3.6 specializes to (4.15)

(−1)|I| |c(I)| =

(−1)|J| |J| ,

J⊆V J free

I⊆V

which in turn specializes to a result of Gordon [Gor97] if c is derived from the convex hull operator in d as in Example 4.1.4. Edelman and Reiner [ER00] showed that in this case either side of (4.15) agrees in absolute value with the number of points in V which are in the interior of conv(V ) if |V | > 1. We continue with a further preliminary proposition. Proposition 4.3.7 [Doh99b] Let {Av }v∈V be a finite family of sets, and let c be a closure operator on V such that for any non-empty and non-closed subset X of V , (4.16) Ax ⊆ Av . x∈X

v ∈X /

Then, for any non-empty subset I of V , Ai = Ai . i∈I

i∈c(I)

show that Proof. Let I ⊆ V , I = ∅. If i∈I Ai = ∅, we are done. Otherwise ω ∈ i∈c(I) Ai for each ω ∈ i∈I Ai . Evidently, for any ω ∈ i∈I Ai , I ⊆ Vω where again Vω is deﬁned as {v ∈ V | ω ∈ Av }. From (4.16) we conclude that Vω is closed and hence, c(I) ⊆ Vω or equivalently, ω ∈ i∈c(I) Ai . 2 We now state and prove the announced generalization of (4.12). Theorem 4.3.8 [Doh99b] Let {Av }v∈V be a finite family of sets and c a closure operator on V such that for any non-empty and non-closed subset X of V , Ax ⊆ Av x∈X

v ∈X /

4.3 Alternative proofs

33

and such that c(∅) = ∅. Then,   Av = (−1)|J|−1 domc (J) χ  Aj  . χ

J∈ ∗ (V ) J closed

v∈V

j∈J

Proof. By the classical inclusion-exclusion identity (1.1) we have χ v∈V Av = |I|−1 g(I) where g(I) := χ i∈I Ai if I = ∅ and g(∅) := 0. By I⊆V (−1) Proposition 4.3.7 and since c(∅) = ∅ we ﬁnd that g = g ◦ c. Now, by applying Proposition 4.3.6, the statement of the theorem immediately follows. 2

4.3.2

Improved inequalities via kernel operators

In the following, we present an elementary proof of the improved Bonferroni inequalities associated with the abstract tube of Theorem 4.2.3 as well as an elementary proof of the fact that these inequalities are at least as sharp as their classical counterparts. The proof given below is new even in the traditional case where k(I) = ∅ for any subset I of V . In this case, it generalizes Garsia and Milne’s bijective proof of the classical inclusion-exclusion principle [GM81, Zei84, Pau86] to an “injective proof” of the classical Bonferroni inequalities. Theorem 4.3.9 [Doh02c] Let {Av }v∈V be a finite family of sets and k a kernel operator on V such that for any non-empty and k-open subset X of V , Ax ⊆ Av . v ∈X /

x∈X

Then, for any r ∈ , χ Av

≥

χ

≤

Av

(−1)|I|−1 χ

I∈ ∗ (V ) k(I)=∅ |I|≤r

v∈V

Ai

(r even),

i∈I

|I|−1

(−1)

χ

I∈ ∗ (V ) k(I)=∅ |I|≤r

v∈V

Ai

(r odd).

i∈I

Proof. Obviously, the preceding two inequalities are equivalent to (4.17) χ Av + χ Ai ≥ χ Ai

I∈ ∗ (V ) k(I)=∅ |I|≤r |I| even

v∈V

(4.18) χ

v∈V

Av

+

I∈ ∗ (V ) k(I)=∅ |I|≤r |I| even

I∈ ∗ (V ) k(I)=∅ |I|≤r |I| odd

i∈I

χ

i∈I

Ai

≤

I∈ ∗ (V ) k(I)=∅ |I|≤r |I| odd

(r even),

i∈I

χ

i∈I

Ai

(r odd).

34

4

Abstract Tubes via Closure and Kernel Operators

For any ω ∈

Av deﬁne Vω := {v ∈ V | ω ∈ Av }, and for any r ∈ , r (ω) := I ∈ (Vω ) k(I) = ∅, |I| ≤ r, |I| even ,

r (ω) := I ∈ (Vω ) k(I) = ∅, |I| ≤ r, |I| odd . v∈V

To prove (4.17) and (4.18), it suﬃces to show that |r (ω)| ≥ | r (ω)| if r is even, and |r (ω)| ≤ | r (ω)| if r is odd. Evidently, Vω is not k-open, so some v ∈ Vω \ k(Vω ) exists. For any I ⊆ Vω it follows that v ∈ / k(I ∪ {v}) since otherwise v ∈ k(I ∪ {v}) ⊆ k(Vω ∪ {v}) = k(Vω ), contradicting v ∈ / k(Vω ). From v∈ / k(I ∪ {v}) and k(I ∪ {v}) ⊆ I ∪ {v} we conclude that k(I ∪ {v}) ⊆ I and hence, k(I ∪ {v}) ⊆ k(I). Therefore, k(I ∪ {v}) = ∅ ⇔ k(I) = ∅. By this, I → I {v}, where denotes symmetric diﬀerence, is an injective mapping from r (ω) into r (ω) if r is even and from r (ω) into r (ω) if r is odd. 2 Theorem 4.3.10 [Doh02c] Let {Av }v∈V be a finite family of sets, and let k and k be kernel operators on V such that k ≤ k with respect to (4.6) or (4.7) and such that for any non-empty and k -open subset X of V , Ax ⊆ Av . v ∈X /

x∈X

Then, for any r ∈ , |I|−1 (−1) χ Ai

I∈ ∗ (V ) k (I)=∅ |I|≤r

≥

(−1)|I|−1 χ

I∈ ∗ (V ) k (I)=∅ |I|≤r

(−1)

≤

Ai

|I|−1

χ

I∈ ∗ (V ) k(I)=∅ |I|≤r

i∈I

Ai

(r even),

i∈I

(−1)|I|−1 χ

I∈ ∗ (V ) k(I)=∅ |I|≤r

i∈I

Ai

(r odd).

i∈I

Proof. Since k ≤ k the preceding two inequalities are equivalent to (4.19) ≥ (r even), χ Ai χ Ai

I∈ ∗ (V ) k(I)=∅ k (I)=∅ |I|≤r |I| even

(4.20)

For any ω ∈

I∈ ∗ (V ) k(I)=∅ k (I)=∅ |I|≤r |I| even

I∈ ∗ (V ) k(I)=∅ k (I)=∅ |I|≤r |I| odd

i∈I

χ

i∈I

Ai

≤

I∈ ∗ (V ) k(I)=∅ k (I)=∅ |I|≤r |I| odd

i∈I

χ

Ai

(r odd).

i∈I

Av deﬁne Vω as above, and for any r ∈ deﬁne ∗r (ω) := I ∈ ∗ (Vω ) k(I) = ∅, k (I) = ∅, |I| ≤ r, |I| even ,

∗r (ω) := I ∈ ∗ (Vω ) k(I) = ∅, k (I) = ∅, |I| ≤ r, |I| odd . v∈V

4.3 Alternative proofs

35

To prove (4.19) and (4.20), we show that |∗r (ω)| ≥ | ∗r (ω)| if r is even, and |∗r (ω)| ≤ | ∗r (ω)| if r is odd. Since Vω is not k -open, some v ∈ Vω \ k (Vω ) exists, and since k ≤ k, v ∈ Vω \ k(Vω ). Similar to the preceding proof, we conclude that k(I {v}) = ∅ ⇔ k(I) = ∅ and k (I {v}) = ∅ ⇔ k (I) = ∅. Hence, I → I {v} is an injective mapping from ∗r (ω) into ∗r (ω) if r is even and from ∗r (ω) into ∗r (ω) if r is odd, thus establishing our claim. 2

4.3.3

Additional proofs of identities

We ﬁnally give two self-contained proofs of the inclusion-exclusion identities associated with the abstract tubes of Corollary 4.2.6 and Corollary 4.2.7. These self-contained proofs only require the traditional form of the inclusion-exclusion principle. The presentation here closely follows [Doh00c] and [Doh99a]. Our ﬁrst proof makes use of a partial ordering relation, which we introduce in the following lemma. Lemma 4.3.11 Let c be a closure operator on some set V . Then, by A ≤c B :⇔ c(A) ∩ B = A

(A, B ⊆ V )

a partial ordering relation on V is defined. Proof. i) ≤c is reﬂexive: A ⊆ c(A) ⇒ c(A) ∩ A = A ⇒ A ≤c A. ii) ≤c is antisymmetric: (A ≤c B) ∧ (B ≤c A) ⇒ (c(A) ∩ B = A) ∧ (c(B) ∩ A = B) ⇒ A = c(A) ∩ B = c(A) ∩ c(B) ∩ A = c(B) ∩ A = B. iii) ≤c is transitive: (A ≤c B) ∧ (B ≤c C) ⇒ (c(A) ∩ B = A) ∧ (c(B) ∩ C = B) ⇒ c(A) ∩ C = c(c(A) ∩ B) ∩ C ⊆ c(A) ∩ c(B) ∩ C = c(A) ∩ B = A ⇒ A ≤c C. 2 Alternative proof of the inclusion-exclusion identity associated with the abstract tube of Corollary 4.2.6. Deﬁne := {I ∈ ∗ (V ) | I ⊇ X for some X ∈ }. Then, by the classical inclusion-exclusion principle, it suﬃces to prove that |I|−1 (4.21) (−1) χ Ai = 0 . I∈

i∈I

We ﬁrst establish that for any Y1 , Y2 ∈ which are ≤c -minimal in , (4.22)

{I ⊆ V | I ≥c Y1 } ∩ {I ⊆ V | I ≥c Y2 } = ∅ ⇒ Y1 = Y2 .

Let I ≥c Y1 and I ≥c Y2 . Then, Y1 = I ∩ c(Y1 ), Y2 = I ∩ c(Y2 ), and hence, c(c(Y1 ) ∩ Y2 ) ∩ Y1 ⊆ Y1 ∩ c(Y2 ) = I ∩ c(Y1 ) ∩ c(Y2 ) = c(Y1 ) ∩ Y2 = c(Y1 ) ∩ Y2 ∩ I ∩ c(Y1 ) ⊆ c(c(Y1 ) ∩ Y2 ) ∩ Y1 . So c(c(Y1 ) ∩ Y2 ) ∩ Y1 = c(Y1 ) ∩ Y2 , which is equivalent to c(Y1 ) ∩ Y2 ≤c Y1 by deﬁnition of ≤c . Since Y1 , Y2 ∈ , there are X1 , X2 ∈ satisfying Y1 ⊇ X1 and Y2 ⊇ X2 . By the requirements of Corollary 4.2.6 is a chain, whence we may assume that c(X1 ) ⊇ c(X2 ). Thus,

36

4

Abstract Tubes via Closure and Kernel Operators

c(Y1 ) ∩ Y2 ⊇ c(X1 ) ∩ X2 ⊇ c(X2 ) ∩ X2 = X2 ∈ and hence, c(Y1 ) ∩ Y2 ∈ . By this and the ≤c -minimality of Y1 in , c(Y1 )∩Y2 = Y1 and hence, Y1 ≤c Y2 . Now, by the ≤c -minimality of Y2 in we conclude that Y1 = Y2 , thus establishing the implication in (4.22). By virtue of (4.22), our claim (4.21) is proved if |I|−1 (4.23) (−1) χ Ai = 0 I≥c Y

i∈I

for each ≤c -minimal Y ∈ . By deﬁnition of ≤c we have   . (−1)|I|−1 χ Ai = (−1)|I|−1 χ  Ay ∩ Ai  I≥c Y

I⊆V \c(Y )

i∈I

y∈Y

i∈I

. where a = b means that a = b or a = −b. By the classical principle of inclusionexclusion we may rewrite the preceding equation as     . (−1)|I|−1 χ Ai = χ  Ay ∩ Ai  − χ  Ay  . I≥c Y

i∈I

i∈c(Y / )

y∈Y

y∈Y

Now, (4.23) (and hence our claim (4.21)) is proved if Ay ⊆ Ai . (4.24) y∈Y

i∈c(Y / )

Since Y ∈ , there is some X ∈ such that Y ⊇ X and hence, c(X) ∩ Y ∈ . It is easy to see that c(c(X) ∩ Y ) ∩ Y = c(X) ∩ Y , whence c(X) ∩ Y ≤c Y and thus c(X) ∩ Y = Y , which is implied by the ≤c -minimality of Y in . Hence, c(Y ) ⊆ c(X) (in fact: c(Y ) = c(X)). This in combination with Y ⊇ X gives Ay ⊆ Ax ⊆ Ai ⊆ Ai , y∈Y

x∈X

i∈c(X) /

i∈c(Y / )

where the second inclusion is justiﬁed by requirement (4.8) of Corollary 4.2.6. Thus, (4.24) holds, and the proof is complete. 2 Alternative proof of the inclusion-exclusion identity associated with the abstract tube of Corollary 4.2.7. Again, it suﬃces to prove (4.21). The idea is to deﬁne a partition of such that for any ∈ , (4.25) (−1)|I|−1 χ Ai = 0 . I∈

i∈I

For any X ∈ deﬁne X + := {v ∈ V | v > max X}. Without loss of generality we may assume that X + = ∅ for any X ∈ . For any M, N ∈ ∗ (V ) deﬁne

4.4 The chordal graph sieve

37

M N if max M < max N or M = N , and let ∗ be a linear # extension of .$ For any I ∈ deﬁne XI := min ∗ {X ∈ | X ⊆ I} and [I] := I \ XI+ , I ∪ XI+ . We now show that := {[I] | I ∈ } is a partition of . Obviously, I ∈ [I] ⊆ for any I ∈ and hence, = I∈ [I]. It remains to show that I ∈ [J] if J ∈ [I]. For J ∈ [I], XJ ∗ XI since J ⊇ XI . By this, max XJ ≤ max XI < min XI+ and therefore, XJ ∩ XI+ = ∅. We conclude that XJ = XJ \ XI+ ⊆ J \ XI+ ⊆ (I ∪ XI+ ) \ XI+ ⊆ I and hence, XI ∗ XJ . From this and XJ ∗ XI it follows that XJ = XI and hence, XJ+ = XI+ . Therefore, J \ XJ+ ⊆ (I ∪ XI+ ) \ XI+ ⊆ I ⊆ (I \ XI+ ) ∪ XI+ ⊆ J ∪ XJ+ , whence I ∈ [J]. We now prove that (4.25) holds for any ∈ . For = [J] we have   . (−1)|I|−1 χ Ai = (−1)|I| χ  Aj ∩ Ai  I∈

I⊆XJ+

i∈I

j∈J\XJ+

i∈I

. where, as in the preceding proof, = means equality up to sign. By applying the classical inclusion-exclusion principle to the right-hand side we obtain   . (−1)|I|−1 χ Ai = χ  Aj ∩ Ax  . I∈

i∈I

j∈J\XJ+

x∈XJ+

From J \ XJ+ ⊇ XJ and precondition (4.9) we conclude that     χ Aj ∩ Ax  ≤ χ  Ax ∩ Ax  = 0 . j∈J\XJ+

x∈XJ+

x∈XJ

x∈XJ+

Now, (4.25) immediately follows from the preceding two equations. 2

4.4

The chordal graph sieve

In Corollary 4.1.14 we saw how chordal graphs give rise to abstract tubes and thus to improved Bonferroni inequalities if some structural requirement on the collection of events applies. In the following theorem this requirement is dropped. The result is a generally-valid inequality where the selection of the intersections in the estimates is determined by a graph. In the literature, such inequalities are referred to as graph sieves [GS96a, GS96b]. The ﬁrst graph sieves were obtained by R´enyi [R´en61] and Galambos [Gal66, Gal72], followed by Hunter [Hun76], Worsley [Wor82], Galambos and Simonelli [GS96a, GS96b], McKee [McK97, McK98] and Buksz´ar and Pr´ekopa [BP01]. The hypertree sieve of Tomescu [Tom86] and related results by Grable [Gra93, Gra94] and Buksz´ar [Buk01] fall into the category of hypergraph sieves, which we do not consider here. We start with the main result of this section—the chordal graph sieve— which associates a Bonferroni-type inequality with any chordal graph G.

38

4

Abstract Tubes via Closure and Kernel Operators

Theorem 4.4.1 [Doh02b] Let {Av }v∈V be a finite family of sets, where V is the vertex-set of a chordal graph G. Then, for any odd r ∈ , (4.26) χ Av ≤ (−1)|I|−1 χ Ai .

I∈ ∗ (V ) I clique of G |I|≤r

v∈V

i∈I

In particular, χ

(4.27)

v∈V

Proof. Let ω ∈

v∈V

(4.28)

Av

≤

(−1)|I|−1 χ

I∈ ∗ (V ) I clique of G

Ai

.

i∈I

Av and Vω := {v ∈ V | ω ∈ Av }. We have to show that 1 ≤ (−1)|I|−1 H

I clique of H 0<|I|≤r

where H runs over all connected components of G[Vω ]. Since each such H is connected and chordal, we can deﬁne a convex geometry (VH , cH ) as in the proof of Corollary 4.1.14, where VH is the vertex-set of H and where for any X ⊆ VH , cH (X) := {z ∈ VH | z is on a chordless path between x and y in H} . x,y∈X

Then, a subset I of VH is cH -free if and only if I is a clique of H. Hence, by Proposition 4.1.8, the abstract simplicial complex H consisting of all nonempty cliques of H is contractible. This in combination with Proposition 3.1.7 shows that the inner sum in (4.28) is at least one, whence (4.28) holds and the proof of (4.26) is complete. To prove (4.27), choose some odd r ≥ |V |. 2 As shown subsequently, the second part of the preceding theorem can be proved in an elementary way by employing the fact that the Euler characteristic of the clique complex of a connected chordal graph is equal to 1.

Alternative proof of (4.27). Let ω ∈ v∈V Av . Obviously, it suﬃces to show that 1 ≤ (4.29) (−1)|I|−1 , H

I clique of H I=∅

where H runs over all connected components of G[Vω ]. Since (4.14) in connection with the convex geometry (VH , cH ) of the preceding proof reveals that the inner sum on the right-hand side of (4.29) is equal to 1, we have the result. 2 Remark. (4.27) can also be proved within the general framework of Galambos and Simonelli [GS96a, GS96b]. Thus, (4.27) originates via [GS96a, Theorem

4.4 The chordal graph sieve

39

II] or [GS96b, Theorem I.2] from the following property of chordal graphs: “for any non-trivial subgraph H of a chordal graph G, the Euler characteristic of the clique complex of H is at least one.” A detailed proof is left as an option to the reader. The following theorem complements the second part of Theorem 4.4.1. Theorem 4.4.2 [Doh02b] Let {Av }v∈V be a finite family of sets, and let G = (V, E) and G = (V, E ) be chordal graphs such that E is a subset of E . Then, |I|−1 |I|−1 (−1) χ Ai ≤ (−1) χ Ai .

I∈ ∗ (V ) I clique of G

I∈ ∗ (V ) I clique of G

i∈I

Proof. Again, let ω ∈

v∈V

Av and Vω := {v ∈ V | ω ∈ Av }. We show that

(4.30)

i∈I

γ(

H )

H

≤

γ(

H)

H

where H and H run over all connected components of G[Vω ] and G [Vω ], respectively, and where γ( H ) and γ( H ) denote the Euler characteristic of the abstract simplicial complexes H and H , which are deﬁned as in the proof of Theorem 4.4.1. Recall from the preceding alternative proof of (4.27) that γ( H ) = γ( H ) = 1, whence (4.30) holds if and only if the number of connected components of G [Vω ] is at most the number of connected components of G[Vω ]. Thus, the result follows from the requirement that E ⊆ E . 2 Remark. Theorem 4.4.1 generalizes several known results. For a complete graph, for instance, we obtain the classical Bonferroni upper bound (1.5), whereas for an edgeless graph we get Boole’s inequality (1.3). In case of a tree we rediscover the following prominent result due to Hunter [Hun76] and Worsley [Wor82]. Corollary 4.4.3 [Hun76, Wor82] Let {Av }v∈V be a finite family of sets, where the indices form the vertices of a tree G = (V, E). Then, (4.31) χ Av ≤ χ(Ai ) − χ(Ai ∩ Aj ) . v∈V

i∈V

{i,j}∈E

Proof. Since trees are chordal, the corollary follows from Theorem 4.4.1. 2 Remarks. 1. By Corollary 4.1.12, the preceding inequality becomes an identity if Ax ∩ Ay ⊆ Az for any x, y ∈ V and any z on the unique path between x and y in G. 2. As observed by Hunter [Hun76] and Worsley [Wor82], the best possible upper bound in (4.31) with respect to some probability measure P is obtained

40

4

Abstract Tubes via Closure and Kernel Operators

by choosing a maximum spanning tree G for the complete weighted graph on V , where each edge {i, j} has weight P (Ai ∩ Aj ). Such a maximum spanning tree can be found eﬃciently by applying the greedy algorithm [Kru56, Pri57]. By induction on m it follows that all graphs of type Km ∗ Ln (the join of a complete graph and an edgeless graph) and Km ∗ Pn (the join of a complete graph and a path) are chordal and thus give rise to inequalities of BonferroniGalambos type via Theorem 4.4.1. Some of these graphs are shown in Figures 4.1–4.3. Remarkably, many well-known and new inequalities can be derived in a uniﬁed and simpliﬁed way by considering these types of graphs. For instance, by considering K1 ∗ Pn−1 (see Figure 4.1) the following inequality is obtained. Corollary 4.4.4 Let A1 , . . . , An be sets. Then, n n n n−1 Ai ≤ χ(Ai ) − χ(Ai−1 ∩ Ai ) − χ(Ai−1 ∩ An ) χ i=1

i=1

i=2

i=2 n−1

+

χ(Ai−1 ∩ Ai ∩ An ) .

i=2

Proof. Consider the graph G = K1 ∗ Pn−1 and apply Theorem 4.4.1. 2 1

2

3

4

n-1

n

Figure 4.1: The graph K1 ∗ Pn−1 . Using K2 ∗Pn−2 (see Figure 4.2) instead of K1 ∗Pn−1 we obtain the following inequality, which by Theorem 4.4.2, is at least as sharp as the preceding one. Corollary 4.4.5 Let A1 , . . . , An be sets. Then, n n n n−1 Ai ≤ χ(Ai ) − χ(Ai−1 ∩ Ai ) − χ(Ai−1 ∩ An ) χ −

i=1 n−2

i=1

i=2

χ(Ai−1 ∩ An−1 ) +

i=2

n−2

i=2 n−2

χ(Ai−1 ∩ Ai ∩ An ) +

i=2

+

n−2 i=1

χ(Ai ∩ An−1 ∩ An ) −

χ(Ai−1 ∩ Ai ∩ An−1 )

i=2 n−2 i=2

χ(Ai−1 ∩ Ai ∩ An−1 ∩ An ) .

4.4 The chordal graph sieve

41

Proof. Consider the graph G = K2 ∗ Pn−2 and apply Theorem 4.4.1. 2 1

2

3

4

n

n-2

n-1

Figure 4.2: The graph K2 ∗ Pn−2 . A well-known inequality is that of Seneta [Sen88]. It easily follows from our chordal graph sieve of Theorem 4.4.1 by considering the graph K2 ∗ Ln−2 (see Figure 4.3). The signiﬁcance of this graph with respect to proving Seneta’s inequality has already been noticed by Galambos and Simonelli [GS96a, GS96b]. Corollary 4.4.6 [Sen88] Let A1 , . . . , An be sets and j, k ∈ {1, . . . , n}. Then, n n n n χ Ai ≤ χ(Ai ) − χ(Ai ∩ Aj ) − χ(Ai ∩ Ak ) i=1

i=1 i=j

i=1

i=1 i=j,k

+

n

χ(Ai ∩ Aj ∩ Ak ) .

i=1 i=j,k

Proof. For j = k deﬁne G = K2 ∗ Ln−2 and apply Theorem 4.4.1. For j = k let G = K1 ∗ Ln−1 be the star having n vertices and apply Theorem 4.4.1 2

j

k

Figure 4.3: The graph K2 ∗ Ln−2 for n = 8. Remarks. For j = k the inequality of Corollary 4.4.6 specializes to n n n χ (4.32) Ai ≤ χ(Ai ) − χ(Ai ∩ Aj ) (j = 1, . . . , n), i=1

i=1

i=1 i=j

42

4

Abstract Tubes via Closure and Kernel Operators

which is due to Kounias [Kou68]. Now, by taking the average over j = 1, . . . , n in Kounias’ inequality we obtain an inequality due to Kwerel [Kwe75b]: n n n 2 χ Ai ≤ χ(Ai ) − χ(Ai ∩ Aj ) . n i,j=1 i=1 i=1 i<j

In a similar way, by taking the average over all distinct j, k = 1, . . . , n in Corollary 4.4.6, another result of Kwerel [Kwe75a] is obtained: n n n n 2n − 3 3 χ Ai ≤ χ(Ai ) − n χ(Ai ∩ Aj ) + n χ(Ai ∩ Aj ∩ Ak ). i=1

2

i=1

2

i,j=1 i<j

i,j,k=1 i<j
We now use Theorem 4.4.1 to deduce a generalization of Kwerel’s inequalities. For even m this generalization coincides with a special case of an inequality due to Galambos and Xu [GX90] (see also Galambos and Simonelli [GS96b]). Theorem 4.4.7 [Doh02b] Let A1 , . . . , An be sets. Then, for m = 0, . . . , n, χ

n

Ai

≤

i=1

m

k−1

(−1)

k=1

m nk − (m + 1)(k − 1) nk χ Ai1 ∩ · · · ∩ Aik m−k+1 k i1 <···
i1 <···
Proof. For m = n the right-hand side of the preceding inequality coincides with the classical inclusion-exclusion formula, whence in the sequel we may assume that m < n. For any subset M of {1, . . . , n} let GM denote the join of the complete graph on M and the edgeless graph on the complement of M . By averaging the right-hand side of (4.27) over all graphs GM with |M | = m we obtain n 1 |I|−1 Ai ≤ n (−1) χ Ai χ m

i=1

=

m+1 k=1

M ⊆{1,...,n} I∈ ∗ ({1,...,n}) I clique of GM |M |=m

k−1

(−1)

I∈ ∗ ({1,...,n}) |I|=k

i∈I

c(n, m, I) n χ m

Ai

,

i∈I

where c(n, m, I) signiﬁes the number of m-subsets M of {1, . . . , n} such that I is a clique of GM . Since I is a clique of GM iff I ⊆ M or |I \ M | = 1, we obtain

n−|I| n−|I| + |I| m−|I|+1 if |I| ≤ m, m−|I| c(n, m, I) = m+1 if |I| = m + 1,

4.4 The chordal graph sieve

43

and hence, χ

n i=1

Ai

≤

m

n−k k−1

(−1)

m−k

n−k + k m−k+1 n m

k=1 m

+ (−1)

χ Ai1 ∩ · · · ∩ Aik

i1 <···
m+1 n

m

i1 <···
χ Ai1 ∩ · · · ∩ Aim+1 .

The result now follows from the preceding inequality and the binomial identity n−k n−k m nk − (m + 1)(k − 1) m−k + k m−k+1 n , = nk m−k+1 m k which is an easy combinatorial exercise. 2

5

Recursive Schemes

This chapter deals with recursive schemes that enable us to eﬃciently compute the probability of a union of events, provided some structural conditions on the events are met. The results may be viewed as a generalization of Shier’s recursive scheme [Shi88, Shi91] for computing the reliability of a coherent system, which in turn generalizes a well-known algorithm of Provan and Ball [PB84] for computing network reliability in pseudopolynomial time. We will come back to Shier’s result in the next chapter when dealing with reliability problems.

5.1

Recursive schemes for semilattices

From Corollary 4.1.11 we now deduce two recursive schemes for the probability of a union. Our results are formulated for upper semilattices only, although in view of Corollary 4.1.20 an even more general formulation would be possible. The following theorem strongly generalizes an important result of Shier [Shi88, Shi91] on system reliability. Theorem 5.1.1 Let {Av }v∈V be a finite family of sets, where V is an upper semilattice such that Ax ∩ Ay ⊆ Ax∨y for any x, y ∈ V . Furthermore, let P be a probability measure on a σ-field containing the events Av , v ∈ V , such that P (Ai2 ∩ · · · ∩ Aik ) > 0 and P Ai1 Ai2 ∩ · · · ∩ Aik = P Ai1 Ai2 for any chain i1 < · · · < ik in V where k > 1. Then, Av = Λ(v) , (5.1) P v∈V

v∈V

where Λ is defined by the following recursive scheme: (5.2)

Λ(v) := P (Av ) −

w>v

K. Dohmen: LNM 1826, pp. 44–46, 2003. c Springer-Verlag Berlin Heidelberg 2003

Λ(w) P (Av |Aw ) .

5.2 Dynamic programming approach

45

Proof. By Corollary 4.1.11 it suﬃces to prove that for any v ∈ V , |I|−1 (5.3) Λ(v) = (−1) P Ai .

I∈ ∗ (V ) I is a chain min I=v

i∈I

We proceed by downward induction on v. If v is maximal in V , then by deﬁnition, Λ(v) = P (Av ), and the statement is proven. For any non-maximal v ∈ V the recursive deﬁnition, the induction hypothesis and the requirements give Λ(v) = P (Av ) − (−1)|I|−1 P Ai P (Av |Aw )

=

=

=

P (Av ) −

P (Av ) +

P (Av ) +

w>v

I∈ ∗ (V ) I is a chain min I=w

w>v

I∈ ∗ (V ) I is a chain min I=w

w>v

I∈ ∗ (V ) I is a chain min I=w

i∈I

|I|−1

(−1)

P

Ai

P

i∈I

Av Ai i∈I

 (−1)|I∪{v}|−1 P 

 Ai 

i∈I∪{v}

|I|−1

(−1)

I∈ ∗ (V ) I is a chain min I=v I={v}

P

Ai

. 2

i∈I

Note that the following theorem is not dual to the preceding one. Theorem 5.1.2 [Doh99d] Let {Av }v∈V be a finite family of sets, where V is an upper semilattice such that Ax ∩ Ay ⊆ Ax∨y for any x, y ∈ V . Furthermore, letP be a probability measure on a σ-field containing the Av , v∈ V, suchthat P Aik−1 ∩ · · · ∩ Ai1 > 0 and P Aik Aik−1 ∩ · · · ∩ Ai1 = P Aik Aik−1 for any ik > · · · > i1 in V where k > 1. Then (5.1) holds where Λ is defined by (5.4) Λ(v) := P (Av ) − Λ(w) P (Av |Aw ) . w
Proof. Theorem 5.1.2 follows by replacing “downward” with “upward”, “min” with “max” and “>” with “<” in the proof of Theorem 5.1.1. 2

5.2

Dynamic programming approach

By applying the technique of dynamic programming, the preceding recursive schemes can be implemented in a quite eﬃcient manner. See Algorithm I for

46

5

Recursive Schemes

a dynamic programming implementation of (5.1) and (5.2). It can easily be adapted to implement (5.1) and (5.4) by replacing line 1 respectively 6 with 1’: Order v1 , . . . , vn ∈ V such that vi < vj ⇒ i < j (i, j = 1, . . . , n) 6’: if vj < vi then

Evidently, Algorithm I has a space complexity of O(|V |) and a time complexity of order |V |2 × T where T is the time needed to compute P (Av ) and P (Av |Aw ). Algorithm I Improved IE algorithm for computing the probability of a union Require: Same requirements

as in Theorem 5.1.1 Ensure: prob = P A v v∈V 1: Find an ordering v1 , . . . , vn of V such that vi < vj ⇒ i > j (i, j = 1, . . . , n) 2: prob ← 0 3: for i = 1 to n do 4: acc ← 0 5: for j = 1 to i − 1 do 6: if vj > vi then 7: acc ← acc + a[j] P (Avi |Avj ) 8: end if 9: end for 10: a[i] ← P (Avi ) − acc 11: prob ← prob + a[i] 12: end for Under the requirements of Theorem 5.1.1, the next theorem shows that the

partial sums of v∈V Λ(v) provide lower bounds on P . Thus, if AlA v v∈V gorithm I is stopped at an arbitrary instant of time, prob provides a lower bound

to P v∈V Av . This is not the case under the requirements of Theorem 5.1.2. Theorem 5.2.1 Under the requirements of Theorem 5.1.1, P Av ≥ Λ(v) for any V ⊆ V . v∈V

v∈V

Proof. For any v ∈ V deﬁne V >v := {i ∈ V | i > v}. Then, by (5.3), we see that Λ(v) = P (Av ) − (−1)|I|−1 P Av ∩ Ai .

I∈ ∗ (V >v ) I is a chain

i∈I

By applying the identity associated with Corollary 4.1.11 with V >v instead of V and integrating the result with respect to the measure P (Av ∩ · ), we obtain % Λ(v) = P (Av ) − P Av ∩ Ai = P Av Ai . i>v

i>v

So Λ(v) is non-negative, whence the result follows from Theorem 5.1.1. 2

6

Reliability Applications

In many practical situations one is interested in the probability that a technical system with randomly failing components is operating. Examples include transportation networks, electrical power systems, pipeline networks and nuclear power plants. Recently, the study of system reliability has received considerable attention from its applicability to computer and telecommunications networks. One of the standard methods in reliability theory is the principle of inclusionexclusion and the associated Bonferroni inequalities. This chapter deals with improvements of this method derived from the results of the preceding chapters. In this way, we rediscover Shier’s semilattice expression and recursive scheme for system reliability [Shi88, Shi91] and establish related inequalities based on abstract tubes. The results are then applied in the more speciﬁc context of network reliability, k-out-of-n systems, consecutive k-out-of-n systems and covering problems, where several results from the literature are rediscovered in a concise way. Examples demonstrate that the new reliability bounds are much sharper than the usual Bonferroni bounds, although less computational eﬀort is needed to compute them. We ﬁnally identify a new class of hypergraphs for which the reliability covering problem can be solved in polynomial time. For an introduction to reliability theory with emphasis on networks we refer to the monographs of Colbourn [Col87] and Shier [Shi91]. For a more general introduction we recommend the monograph of Aven and Jensen [AJ99].

6.1 6.1.1

System reliability Terminology

The following deﬁnition essentially goes back to Esary and Proschan [EP63]. Definition 6.1.1 A coherent system or—to be more precise—a coherent binary system is a pair Σ = (E, φ) where E is a ﬁnite set and φ : (E) → {0, 1} is a function satisfying φ(∅) = 0, φ(E) = 1 and X ⊆ Y ⇒ φ(X) ≤ φ(Y ) for any X, Y ⊆ E. We refer to E and φ as the component set resp. the structure function of the system. Each component e ∈ E assumes randomly and independently K. Dohmen: LNM 1826, pp. 47–81, 2003. c Springer-Verlag Berlin Heidelberg 2003

48

6

Reliability Applications

one of two states, operating or failing, with probabilities pe and qe = 1 − pe , respectively. The set of operating components of Σ is referred to as the state of Σ. Σ is said to be operating if φ applied to its state gives 1; otherwise, Σ is said to be failing. The probability for the event that Σ is operating is referred to as the reliability of Σ. We use the abbreviation RelΣ (p), where p = (pe )e∈E , to denote the reliability of Σ for given component operation probabilities (pe )e∈E . Remark. Note that by the preceding deﬁnition the components of a coherent system fail in a statistically independent fashion. Of course, this deﬁnition could be relaxed by allowing dependent component failures, and indeed most of our inclusion-exclusion expressions for system and network reliability can easily be adapted to this more general setting. For ease of presentation, however, we prefer to use the more restrictive deﬁnition. A generalization to dependent component failures is (whenever possible) left as an option to the reader. The following notions are of vital importance in reliability analysis. Definition 6.1.2 Let Σ = (E, φ) be a coherent binary system. A minpath of Σ = (E, φ) is a minimal set P ⊆ E such that φ(P ) = 1 (that is, φ(P ) = 1 and φ(Q) = 0 for any Q ⊂ P ). A mincut of Σ is a minimal set C ⊆ E such that φ(E \ C) = 0 (that is, φ(E \ C) = 0 and φ(E \ D) = 1 for any D ⊂ C). Thus, with denoting the set of minpaths resp. mincuts, {F operates} resp. 1 − RelΣ (p) = P {F fails} , RelΣ (p) = P F ∈

F ∈

where P denotes the probability measure on the set of system states, and where {e operates}; {F fails} := {e fails} {F operates} := e∈F

e∈F

for any F ∈ . In the following, ( ) again denotes the order complex of .

6.1.2

Abstract tubes for system reliability

The following theorem is of vital importance in this chapter. For a generalization of this theorem from semilattices to convex geometries, see [Doh03]. Theorem 6.1.3 [Doh99d] Let Σ = (E, φ) be a coherent binary system, whose set of minpaths resp. mincuts is given the structure of a lower semilattice such that X ∧ Y ⊆ X ∪ Y for any X, Y ∈ . Then, & ' & ' {F operates} F ∈ , ( ) resp. {F fails} F ∈ , ( ) is an abstract tube.

6.1 System reliability

49

Proof. Apply Corollary 4.1.11 (dualized) with V := and AF := {F operates} resp. AF := {F fails} for any F ∈ . 2 Remark. The improved inclusion-exclusion identity associated with the abstract tube of the preceding theorem is due to Shier [Shi88, Shi91]. Subsequently, the improved inequalities associated with this abstract tube are explicitly stated. Corollary 6.1.4 [Doh99c] Let Σ = (E, φ) be a coherent system, whose set of minpaths resp. mincuts is a lower semilattice such that X ∧ Y ⊆ X ∪ Y for any X, Y ∈ , and let r ∈ . Then, in case that denotes the set of minpaths, ( (−1)||−1 pe (r even), RelΣ (p) ≥ ∈() ||≤r

RelΣ (p) ≤

e∈

Ë

e∈

Ë

(

(−1)||−1

∈() ||≤r

pe

(r odd),

and in case that denotes the set of mincuts, 1 − RelΣ (p) ≥ 1 − RelΣ (p) ≤

(−1)||−1

∈() ||≤r

∈() ||≤r

(−1)||−1

(

Ë

qe

(r even),

e∈

Ë

qe

(r odd),

e∈

(

where in both cases p = (pe )e∈E ∈ [0, 1]E and qe = 1 − pe for any e ∈ E. Proof. The result directly follows from Theorem 6.1.3 and Theorem 3.1.9. 2 Remarks. In view of Theorem 3.1.10 the inequalities of Corollary 6.1.4 are at least as sharp as their classical counterparts, which are obtained if is a chain, that is, if X ∧ Y = X or X ∧ Y = Y for any X, Y ∈ . Note that Corollary)6.1.4 can easily) be generalized to dependent component failures by replacing e∈Ë pe resp. e∈Ë qe with the probability that all

components in work resp. fail.

6.1.3

Shier’s pseudopolynomial algorithm

The following theorem, due to Shier [Shi88, Shi91], generalizes an important algorithm of Provan and Ball [PB84, BP87] for computing network reliability in pseudopolynomial time. The theorem requires the following notion of convexity: Definition 6.1.5 A subset X of a partially ordered set V is convex if [x, y] ⊆ X for any x, y ∈ X, where [x, y] denotes the interval {z ∈ V | x ≤ z ≤ y}.

50

6

Reliability Applications

Theorem 6.1.6 [Shi88, Shi91] Let Σ = (E, φ) be a coherent binary system, whose set of minpaths resp. mincuts is a lower semilattice such that X ∧ Y ⊆ X ∪ Y for any X, Y ∈ and {F ∈ | e ∈ F } is convex for any e ∈ E. Then, (6.1) Λ(F, p) resp. 1 − RelΣ (p) = Λ(F, q) , RelΣ (p) = F ∈

F ∈

where q = 1 − p ∈ [0, 1]E , and where in both cases Λ is defined recursively by ( ( xe − Λ(G, x) xe ; x = (xe )e∈E ∈ [0, 1]E . (6.2) Λ(F, x) := e∈F

e∈F \G

G
Based on Theorem 5.1.1 we now establish a new and simpliﬁed proof of Theorem 6.1.6, which does not make use of the disjoint products technique. Proof. We apply Theorem 5.1.1 (dualized) with V := and AF := {F operates} resp. AF := {F fails}. By the assumptions, AX ∩AY ⊆ AX∧Y for any X, Y ∈ . Deﬁne Be := {e operates} resp. Be := {e fails} for any e ∈ E. Then, for any F ∈ , AF = e∈F Be and thus, by Theorem 5.1.1, we have to check that = P Bi Bi ∩ · · · ∩ Bi Bi Bi P i∈I1

i∈I2

i∈Ik

i∈I1

i∈I2

for any chain I1 > · · · > Ik in where k > 1 and where P again denotes the induced probability measure on the set of system states. Since the components of the system are assumed to operate and fail independently, we ﬁnd that   P = P Bi Bi ∩ · · · ∩ Bi Bi  . i∈I1

i∈I2

i∈Ik

i∈I1 \(I2 ∪···∪Ik )

By the convexity assumption, I1 \ (I2 ∪ · · · ∪ Ik ) = I1 \ I2 and therefore,         P Bi Bi Bi Bi , = P = P i∈I1 \(I2 ∪···∪Ik )

i∈I1 \I2

i∈I1

i∈I2

where the last equals sign again follows from the independence assumption. 2 Remarks. In view of Theorem 5.1.2, replacing “<” with “>” in (6.2) would result in a diﬀerent recursive scheme for RelΣ (p). By suitably adapting Algorithm I, we obtain Shier’s dynamic programming solution to (6.1) and (6.2), which is restated as Algorithm II. As pointed out by Shier [Shi88, Shi91], this algorithm has a space complexity of O(| |) and a time complexity of O(|E| × | |2), since there are at most O(| |2 ) products to be calculated in line 7 and the calculation of each requires at most O(|E|) work. Thus, the algorithm is pseudopolynomial, that is, its running time is bounded by a polynomial in the number of

6.1 System reliability

51

components and the number of minpaths resp. mincuts. The classical inclusionexclusion method for the same problem has a time complexity of O(|E| × 2||). While the classical inclusion-exclusion method and the improved Bonferroni inequalities provided by Corollary 6.1.4 can easily be adapted to work for statistically dependent component failures, Theorem 6.1.6 strongly relies on the independence assumption. Algorithm II Pseudopolynomial algorithm for computing system reliability Require: Same requirements as in Theorem 6.1.6; x = (xe )e∈E ∈ [0, 1]E Ensure: prob = F ∈ Λ(F, x) 1: Find an ordering F1 , . . . , Fn of such that Fi < Fj ⇒ i < j (i, j = 1, . . . , n) 2: prob ← 0 3: for i = 1 to n do 4: acc ← 0 5: for j = 1 to i − 1 do 6: if Fj < Fi then ) 7: acc ← acc + a[j] e∈Fi \Fj xe 8: end if 9: end for ) 10: a[i] ← e∈Fi xe − acc 11: prob ← prob + a[i] 12: end for In the same way as Theorem 6.1.6 follows from Theorem 5.1.1, the following inequalities follow from Theorem 5.2.1. These inequalities ensure that during execution of Algorithm II, prob provides a lower bound to the desired value. Theorem 6.1.7 Under the requirements of Theorem 6.1.6, RelΣ (p) ≥ Λ(F, p) resp. 1 − RelΣ (p) ≥ Λ(F, q) F ∈

F ∈

for any subset of , where q = 1 − p and Λ is defined recursively as in (6.2). Remark. Without modifying the proofs, the preceding theorems can be generalized by imposing the somewhat weaker requirement that is an extended set of minpaths resp. mincuts of Σ. These concepts are introduced subsequently. Definition 6.1.8 An extended set of minpaths of a coherent binary system Σ is an upper set of the set of minpaths of Σ such that any F ∈ is an upper set of some minpath of Σ. An extended set of mincuts of Σ is an upper set of the set of mincuts of Σ such that any F ∈ is an upper set of some mincut of Σ. We will make use of these notions when considering particular reliability systems in Section 6.2.

52

6.1.4

6

Reliability Applications

Inclusion-exclusion and domination theory

The following concept of domination was introduced by Satyanarayana and Prabhakar [SP78] in the speciﬁc context of source-to-terminal reliability. Definition 6.1.9 Let Σ = (E, φ) be a coherent binary system, whose set of minpaths resp. mincuts is denoted by . An -formation of a subset X of E is

any subset of such that = X. An -formation of X is odd resp. even if || is odd resp. even. The -domination of X, dom (X), is the number of odd -formations of X minus the number of even -formations of X. The subsequent theorem provides an improved inclusion-exclusion formula for the reliability of a coherent system based on the concept of domination. Note that this formula contains no cancelling terms, and in this sense it is best possible among all inclusion-exclusion expansions. However, apart from some particular measures associated with directed networks [SP78, SH81, Sat82] and consecutively connected systems [KP89, SM91] the determination of dom (X) is a tiresome task and in fact amounts to computing the M¨ obius function of a lattice. Theorem 6.1.10 [SP78] Let Σ = (E, φ) be a coherent binary system, whose set of minpaths resp. mincuts is denoted by . Then, ( RelΣ (p) = dom (X) pe , X∈∗ (E)

e∈X

(

respectively 1 − RelΣ (p) =

dom (X)

X∈∗ (E)

qe ,

e∈X

where p = (pe )e∈E ∈ [0, 1]E and qe = 1 − pe for any e ∈ E. Proof. Theorem 6.1.10 is proved by applying the classical inclusion-exclusion principle. 2 ∗ Definition 6.1.11 For any set of minpaths or mincuts we use todenote ∗ the set of all unions of sets in including ∅, that is, := ⊆ .

Clearly, ∗ is a lattice with respect to the usual inclusion order. The following interpretation of dom (X) in terms of the M¨ obius function of ∗ is due to Manthei [Man90, Man91] and proved here in a new and simpliﬁed way without using Rota’s crosscut theorem (see p. 96). We ﬁrst deﬁne the M¨obius function. Definition 6.1.12 The M¨ obius function of a ﬁnite partially ordered set P with least element ˆ 0 is the unique -valued function µP on P such that for any x ∈ P , (6.3) µP (y) = δˆ0x , y≤x

where δ is the usual Kronecker delta.

6.2 Special reliability systems

53

Theorem 6.1.13 [Man90, Man91] Let Σ = (E, φ) be a coherent binary system, whose set of minpaths resp. mincuts is denoted by . Then, for any X ∈ ∗, dom (X) = −µ∗ (X) . Proof. By the deﬁnition of the M¨ obius function it suﬃces to prove that

−1 if X = ∅, dom (Y ) = 0 otherwise, Y ∈∗ Y ⊆X

which is clear if X = ∅. Otherwise, |X := {F ∈ | F ⊆ X} = ∅ and hence, dom (Y ) = (−1)||−1 = (−1)||−1 = (−1)||−1 = 0 , Y ∈∗ Y ∈∗ Ë⊆ ⊆|X Ë⊆⊆X Y ⊆X Y ⊆X =Y which gives the result. 2 The domination concepts of Deﬁnition 4.3.4 and 6.1.9 are related as follows. Theorem 6.1.14 [Doh99b] Let Σ = (E, φ) be a coherent binary system, whose set of minpaths resp. mincuts

is denoted by , and let c denote the closure operator → F ∈ F ⊆ on . Then, for any X ∈ ∗ , dom (X) = (−1)||X|−1domc ( |X) , where |X := {F ∈ | F ⊆ X}. Proof. Evidently, |X is c -closed. Hence, by Proposition 4.3.5, we obtain (−1)||−1 = (−1)||X|−1domc ( |X) . c

⊆|X ()=|X

Now, c () = |X if and only if is an -formation of X. Thus, the left-hand side of the preceding equation coincides with the -domination of X. 2

6.2 6.2.1

Special reliability systems Communications networks

Throughout this section, a network is viewed as a ﬁnite graph or digraph, whose nodes or vertices are perfectly reliable and whose edges, which are either undirected or directed, are subject to random and independent failure, where all failure probabilities are assumed to be known in advance. The reader should keep in mind that the improved inclusion-exclusion identities and Bonferroni inequalities in this section can easily be generalized to dependent edge failures

54

6

Reliability Applications

provided the joint probability distribution of the edge failures is known. The general objective is to assess the overall reliability of the network relative to a given reliability measure. Some reliability measures are reviewed subsequently. The source-to-terminal reliability or two-terminal reliability of a network is the probability that a message can pass from a distinguished source node s to a distinguished terminal node t along a path of operating edges. Here and subsequently, paths are considered as directed if the network is directed and as undirected if the network is undirected, and it is assumed that there is always an operating path from each node to itself. The problem of computing the sourceto-terminal reliability for a distinguished source node s and a distinguished terminal node t is usually referred to as the s, t-connectedness problem. It has been studied extensively in the literature (see e.g., [Shi91] for a comprehensive account). An appropriate model for the source-to-terminal measure is a coherent binary system Σ = (E, φ), where E is the edge-set of the network and φ(X) = 1 if and only if X contains the edges of an s, t-path. Evidently, the minpaths and mincuts of the system correspond to the s, t-paths and s, t-cutsets (= minimal sets of edges whose removal disconnects s from t) of the network, respectively. As a generalization of the s, t-connectedness problem, the s, T -connectedness problem asks for the probability that a message can be sent from a distinguished source node s to each node of some speciﬁed set T along a path of operating edges. This probability is usually referred to as the source-to-T -terminal reliability. An appropriate model for dealing with this network reliability measure is a coherent binary system Σ = (E, φ), where E is the edge-set of the network and φ(X) = 1 if and only if X contains the edges of an s, T -tree (= minimal subnetwork containing an s, t-path for all t ∈ T ). In this case, the minpaths of the system correspond to the s, T -trees of the network and the mincuts to its s, T -cutsets (= minimal sets of edges whose removal disconnects s from at least one node in T ). A huge literature exists on the all-terminal reliability (see e.g., [Col87] for an extensive account), which expresses the probability that a message can be sent between any two nodes of the network along a path of operating edges. In the case of an undirected network, for instance, the appropriate model is a coherent binary system Σ = (E, φ), where E is the edge-set of the network and φ(X) = 1 if and only if the subnetwork induced by X is connected. Thus, in the undirected case the minpaths correspond to the spanning trees of the network and the mincuts to its cutsets (= minimal sets of edges whose removal disconnects the network). The methods of the preceding section require a generation of the minpaths resp. mincuts as well as an appropriate semilattice structure on the set of these objects. For most relevant network reliability measures these key objects can be generated quite eﬃciently, that is, their generation time grows only polynomially (or even linearly) with their number. For instance, Tsukiyama et al. [TSOA80] devised an eﬃcient algorithm for generating all s, t-cutsets in an undirected network that has a time complexity of only O((n + m)cst ) where n, m and cst denote the number of nodes, edges and s, t-cutsets of the network, respectively. More recently, Provan and Shier [PS96] established a unifying paradigm for

6.2 Special reliability systems

55

generating all s, t-cutsets and several other classes of cuts in directed and undirected networks that exhibits a worst-case time complexity growing only linearly with the number of objects generated. Eﬃcient algorithms for generating s, tpaths are provided by Read and Tarjan [RT75] as well as Colbourn [Col87]. Shier [Shi91] describes several algebraic enumeration techniques for generating these key objects based on a symbolic version of the Gauss-Jordan algorithm. Unfortunately, the number of key objects can grow exponentially with the size (= number of nodes and edges) of the network, whence the methods of the preceding section exhibit an exponential time behaviour in the worst case. On the other hand, the computation of nearly all relevant network reliability measures is known to be #P -hard [Val79, Bal86] (see also Garey and Johnson [GJ79] for notions of computational complexity), whence a polynomial time algorithm (that is, an algorithm whose time complexity is bounded by a polynomial in the size of the network) for exactly computing any of these measures is unlikely to exist. Fortunately, though, in many practical situations the networks are sparse and thus do not have too many minpaths or mincuts. In view of this and the fact that the method of inclusion-exclusion is a standard method in system and network reliability analysis, it is reasonable to investigate improvements of this method. Now, in order to apply the results of the preceding section to a network, an appropriate partial ordering relation on the set of minpaths resp. mincuts (or, more generally, on an extended set of minpaths resp. mincuts) must be imposed. The following partial ordering relations, which are adopted from Shier [Shi88, Shi91], are appropriate for dealing with the source-to-terminal reliability measure. I. For edge-sets X and Y of s, t-paths in a planar network deﬁne (6.4)

X≤Y

:⇔

X lies below Y .

Of course, this partial ordering relation depends on a speciﬁc drawing of the network in the plane with no edges crossing and with s and t lying on the boundary of the exterior region. Figure 6.1 clariﬁes this concept. II. For s, t-cutsets X and Y of an arbitrary network deﬁne (6.5)

X ≤Y

:⇔

Ns (X) ⊆ Ns (Y ) ,

where Ns (X) is the set of nodes reachable from s after removing X. It is easy to see that these partial ordering relations induce a lattice structure, where X ∧ Y and X ∨ Y are included in X ∪ Y . Moreover, both partial ordering relations satisfy the convexity requirement of Theorem 6.1.6. We thus conclude that Theorems 6.1.3–6.1.7 can be applied to networks whose s, t-paths resp. s, t-cutsets are ordered as in (6.4) resp. (6.5). For s, t-paths in planar networks, ordered as in (6.4), the improved inclusion-exclusion identity associated with the abstract tube of Theorem 6.1.3 is due to Shier [Shi88, Shi91], while the associated improved Bonferroni inequalities are due to the author [Doh99d]. For s, t-cutsets of arbitrary networks, ordered as in (6.5), the improved inclusionexclusion identity associated with the abstract tube of Theorem 6.1.3 coincides

56

6

Reliability Applications

with Buzacott’s node partition formula [BC84, Buz87], whereas the improved inequalities were ﬁrst established in [Doh99d]. For complete undirected networks on n nodes the classical and improved inclusion-exclusion identity for two-terminal reliability based on the s, t-cutsets of the network are compared in [Vog99] for some small values of n, see Table 6.2 for details. In [Vog99] similar results were reported for random undirected networks. We further remark that Theorem 6.1.6, when applied to s, t-cutsets, specializes to a wellknown result of Provan and Ball [PB84], whereas for s, t-paths it is again due to Shier [Shi88, Shi91]. For further informations on these partial ordering relations, the reader is referred to Shier [Shi88, Shi91]. X

Y

s

t inf(X,Y)

Figure 6.1: Two s, t-paths and their inﬁmum [Shi91].

n

number of terms n−2 classical 22 improved

3 4 5 6 7 8

4 16 256 65535 4294967296 18446744073709551616

4 12 52 300 2164 18732

Figure 6.2: Classical and improved inclusion-exclusion [Vog99]. A partial ordering relation on the set of s, t-paths or s, T -cutsets of a directed network satisfying the requirements of Theorem 6.1.6 is diﬃcult to ﬁnd, since such a partial ordering relation gives rise to a pseudopolynomial algorithm for computing two-terminal resp. source-to-T -terminal reliability. Here, pseudopolynomial means that the worst-case time complexity of the algorithm is bounded by a polynomial in the number of s, t-paths resp. s, T -cutsets of the network. However, by a well-known result due to Provan and Ball [PB84], such an algorithm cannot exist unless the complexity classes P and N P coincide. For complete networks, however, we can establish a partial ordering relation on the set of s, T -cutsets that satisﬁes the requirements of Theorem 6.1.6 by deﬁning X ≤ Y as in (6.5) for any s, T -cutsets X and Y of the network. Indeed, this partial ordering relation induces a lower semilattice such that X∧Y ⊆ X∪Y ; see [Doh98] for details. The convexity requirement is easily veriﬁed. We remark that for s, t-cutsets of arbitrary networks and s, T -cutsets of

6.2 Special reliability systems

57

complete networks, the recursive scheme of Theorem 6.1.6 goes back to Ball and Provan [PB84, BP87]. As in [Doh99c], we now establish a partial ordering relation, which is appropriate for dealing with the all-terminal reliability of an undirected network and which leads to an improvement upon the classical inclusion-exclusion identities and inequalities if the network is suﬃciently dense. To this end, let G be an undirected network with vertex-set V , and for any non-empty proper subset W of V let W be the quasi-cut associated with W , that is, the set of edges linking some node in W to some node in V \ W . Then, := {W | ∅ = W ⊂ V } is an extended set of mincuts of the coherent system associated with G and the all-terminal reliability measure. Note that coincides with the set of cutsets of the network if and only if the network is complete. Now, ﬁx some v ∈ V , and for any non-empty proper subsets W1 and W2 of V containing v deﬁne W1 ≤ W2 :⇔ W1 ⊆ W2 .

(6.6)

In this way, becomes a lower semilattice where W1 ∧ W2 = W1 ∩ W2 ⊆ W1 ∪ W2 for any non-empty proper subsets W1 and W2 of V containing v. For complete undirected networks, whose cutsets are ordered as in (6.6), the improved identity associated with the abstract tube of Theorem 6.1.3 coincides with a particular case of Buzacott’s node partition formula [BC84]. As noted by Buzacott [BC84], the formula contains no cancelling terms in this case and thus is best possible among all cutset-based inclusion-exclusion expansions. Example 6.2.1 [Doh99c, Doh99d] Consider the network in Figure 6.3. We are interested in bounds for its two-terminal reliability with respect to s and t. For simplicity, assume that all edges fail independently with probability q = 1 − p. Let’s ﬁrst consider the classical and improved approach based on

6

1

4 2

s

7

t

5 3

8

Figure 6.3: A sample network with terminal nodes s and t. the s, t-paths of the network, which are assumed to be partially ordered as proposed in (6.4). The Hasse diagram corresponding to this partial ordering relation is shown in Figure 6.4(a), and the corresponding bounds (both classical and improved ones) are listed in Table 6.1 together with the number of sets

58

6

Reliability Applications

16

147

1458

678

246

27

258

3567

24568

1478

2346

13457

1258

3546

357

38

123

(a) Hasse diagram of s, t-paths

(b) Hasse diagram of s, t-cutsets

Figure 6.4: Hasse diagrams for the network in Figure 6.3.

inspected during the computation of each bound. Note that the classical bounds come from the classical Bonferroni inequalities, whereas the improved bounds are those of Corollary 6.1.4. Similarly, the s, t-cutsets of the network, which are assumed to be partially ordered as in (6.5), give rise to the Hasse diagram in Figure 6.4(b) and the bounds in Table 6.2. Note that in Table 6.1 even and odd values of r correspond to lower and upper bounds on the reliability of the network, respectively, whereas in Table 6.2 the correspondence is vice versa. In each case, the last bound represents the exact reliability of the network. As expected, the improved bounds involve much fewer terms (sets) than their classical counterparts, although they are much closer to the exact reliability. A numerical comparison of classical and improved bounds is shown in Tables 6.3 and 6.4, and in Figures 6.6 and 6.7 some of these bounds are plotted. To illustrate the bounds provided by Theorem 6.1.7, we consider the Hasse diagram of s, t-paths in Figure 6.4(a). (Of course, we could also employ the s, t-cutsets.) Straightforward application of the recursive scheme (6.2) gives Λ(38) = p2 , Λ(258) = p3 − p4 , Λ(357) = p3 − p4 , Λ(1458) = p4 − 2p5 + p6 ,

Λ(27) = p2 − 3p4 + 2p5 , Λ(3546) = p4 − 2p5 + p6 , Λ(147) = p3 − p4 − 3p5 + 5p6 − 2p7 , Λ(246) = p3 − p4 − 3p5 + 5p6 − 2p7 ,

where the expression Λ(e1 . . . en ) is used as an abbreviation for Λ({e1 , . . . , en }, p). Now, by virtue of Theorem 6.1.7, these Λ-values give rise to lower bounds for the two-terminal reliability of our sample network in Figure 6.3. We thus obtain

6.2 Special reliability systems

59

e.g. the following lower bounds, which are plotted in Figure 6.8: c0 := Λ(38) = p2 , c1 := c0 + Λ(258) + Λ(357) = p2 + 2p3 − 2p4 , c2 := c1 + Λ(1458) + Λ(27) + Λ(3546) = 2p2 + 2p3 − 3p4 − 2p5 + 2p6 , c3 := c2 + Λ(147) + Λ(246) = 2p2 + 4p3 − 5p4 − 8p5 + 12p6 − 4p7 . In a similar way, bounds for the all-terminal reliability of the network in Figure 6.3 are obtained by employing the cutsets and quasi-cuts of the network. The Hasse diagram of the quasi-cuts, which are assumed to be partially ordered as in (6.6), is shown in Figure 6.5, and the corresponding bounds can be read from Table 6.5 together with the number of sets inspected during the computation of each bound. Here, the classical bounds are the classical Bonferroni bounds based on the cutsets of the network, whereas the improved bounds are obtained by applying Corollary 6.1.4 to the quasi-cuts of the network. Note that in Table 6.5 even and odd values of r correspond to upper and lower bounds, respectively. Again, we observe that the improved bounds are much sharper than the classical ones. Some numerical values of the bounds in Table 6.5 are shown in Table 6.6, and in Figure 6.9 some of these bounds are plotted.

23478

358

678

2457

146

3567

24568

1478

134568

2346

13457

1258

123678

12567

123

Figure 6.5: Hasse diagram of quasi-cuts of the network in Figure 6.3.

4p3

2p4 9 45 129 255 381 465 501 510 511

sets 4p3

2p4

+ + 3p2 + 4p3 − 9p4 − 14p5 3p2 + 4p3 − 9p4 − 10p5 3p2 + 4p3 − 9p4 − 10p5 3p2 + 4p3 − 9p4 − 10p5

3p2 − 2p6 + 27p6 + 4p7 + 27p6 − 18p7 − 2p8 + 27p6 − 18p7 + 4p8

improved bounds a∗r 9 36 73 97 103

sets

4q 4

2q 5

1− − − 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6

2q 3 + 10q 7 − 22q 7 − 14q 7 − 14q 7 − 14q 7 − 14q 7 − 14q 7

classical bounds br + q8 − 23q 8 + 39q 8 − 17q 8 + 11q 8 + 3q 8 + 4q 8

9 37 93 163 219 247 255 256

sets 4q 4

2q 5

1− − − 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 + 2q 7 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 − 14q 7 − 2q 8 1 − 2q 3 − 4q 4 + 2q 5 + 13q 6 − 14q 7 + 4q 8

2q 3

improved bounds b∗r

9 28 46 52

sets

Table 6.2: Cutset-based bounds for the two-terminal reliability of the network in Figure 6.3.

1 2 3 4 5 6 7 8

r

Table 6.1: Path-based bounds for the two-terminal reliability of the network in Figure 6.3.

+ + 3p2 + 4p3 − 9p4 − 16p5 − 9p6 3p2 + 4p3 − 9p4 − 8p5 + 34p6 + 30p7 + 3p8 3p2 + 4p3 − 9p4 − 10p5 + 27p6 − 50p7 − 34p8 3p2 + 4p3 − 9p4 − 10p5 + 27p6 − 12p7 + 54p8 3p2 + 4p3 − 9p4 − 10p5 + 27p6 − 18p7 − 24p8 3p2 + 4p3 − 9p4 − 10p5 + 27p6 − 18p7 + 12p8 3p2 + 4p3 − 9p4 − 10p5 + 27p6 − 18p7 + 3p8 3p2 + 4p3 − 9p4 − 10p5 + 27p6 − 18p7 + 4p8

3p2

classical bounds ar

6

1 2 3 4 5 6 7 8 9

r

60 Reliability Applications

6.2 Special reliability systems

.

1 0.8 0.6

Rel a*4 a4 a*3 a3

0.4 0.2 p 0.2

0.4

0.6

0.8

1

Figure 6.6: Some path-based bounds of Table 6.1.

1 0.8 0.6 0.4

Rel b*3 b3 b*2 b2

0.2 q 0.2

0.4

0.6

0.8

1

Figure 6.7: Some cutset-based bounds of Table 6.2.

61

62

6

Reliability Applications

p

a2

a∗2

a4

a∗4

a∗5 †

a5

a∗3

a3

0.1 0.2 0.3 0.4 0.5

0.033 0.132 0.260 0.305 0.047

0.033 0.133 0.270 0.354 0.219

0.033 0.135 0.287 0.410 0.273

0.033 0.136 0.296 0.483 0.648

0.033 0.136 0.297 0.487 0.672

0.033 0.136 0.301 0.530 0.914

0.033 0.136 0.301 0.520 0.828

0.033 0.138 0.317 0.614 1.215

† two-terminal

reliability

Table 6.3: Numerical values of the bounds in Table 6.1.

q

b3

b∗3

b5

b∗4 †

b4

b∗2

b2

0.1 0.2 0.3 0.4 0.5

0.998 0.979 0.922 0.792 0.504

0.998 0.979 0.925 0.819 0.648

0.998 0.979 0.924 0.809 0.590

0.998 0.979 0.925 0.823 0.672

0.998 0.979 0.927 0.846 0.809

0.998 0.979 0.928 0.847 0.781

0.998 0.979 0.930 0.860 0.848

† two-terminal

reliability

Table 6.4: Numerical values of the bounds in Table 6.2.

1 0.8 0.6

Rel c3 c2 c1 c0

0.4 0.2 p 0.2

0.4

0.6

0.8

1

Figure 6.8: Path-based bounds c0 , c1 , c2 , c3 .

1− 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3 1 − 4q 3

4q 3

− − 5q 4 − 5q 4 − 5q 4 − 5q 4 − 5q 4 − 5q 4 − 5q 4 − 5q 4 − 5q 4 − 5q 4 − 5q 4 − 5q 4

5q 4

− + 8q 5 + 4q 5 + 4q 5 + 4q 5 + 4q 5 + 4q 5 + 4q 5 + 4q 5 + 4q 5 + 4q 5 + 4q 5 + 4q 5

4q 5

+ 44q 6 + 30q 6 + 30q 6 + 30q 6 + 30q 6 + 30q 6 + 30q 6 + 30q 6 + 30q 6 + 30q 6 + 30q 6 + 30q 6

+ 20q 7 − 160q 7 + 40q 7 − 68q 7 − 36q 7 − 40q 7 − 40q 7 − 40q 7 − 40q 7 − 40q 7 − 40q 7 − 40q 7

classical bounds dr + 2q 8 − 86q 8 + 429q 8 − 750q 8 + 934q 8 − 778q 8 + 509q 8 − 206q 8 + 80q 8 + 2q 8 + 15q 8 + 14q 8

13 91 377 1092 2379 4095 5811 7098 7813 8099 8177 8190 8191

sets 1− 1 − 4q 3 1 − 4q 3 1 − 4q 3

4q 3 − − 5q 4 − 5q 4 − 5q 4

5q 4 − + 4q 5 + 4q 5 + 4q 5

4q 5 − + 30q 6 + 8q 7 + 2q 8 + 30q 6 − 40q 7 − 10q 8 + 30q 6 − 40q 7 + 14q 8

2q 6

improved bounds d∗r 15 65 125 149

sets

0.995 0.959 0.842 0.575 0.063

0.1 0.2 0.3 0.4 0.5

d3 0.996 0.961 0.842 0.461 -0.805

reliability

0.995 0.959 0.840 0.567 0.031

d∗1 0.996 0.963 0.874 0.708 0.430

d∗3 0.996 0.963 0.875 0.723 0.523

d∗4 † 0.996 0.965 0.920 1.127 2.770

d4

0.996 0.963 0.885 0.794 0.852

d∗2

0.996 0.966 0.908 0.912 1.289

d2

Table 6.6: Numerical values of the bounds in Table 6.5.

† all-terminal

d1

q

Table 6.5: Cutset-based bounds for the all-terminal reliability of the network in Figure 6.3.

1 2 3 4 5 6 7 8 9 10 11 12 13

r

6.2 Special reliability systems 63

64

6

Reliability Applications

1 Rel

0.8

d2 0.6

d*2

0.4

d3 d*3

0.2

0.2

0.4

0.6

0.8

1

q

Figure 6.9: A plot of some of the bounds in Table 6.5.

Comparison with best linear Bonferroni bounds So far, we saw that our abstract tube bounds for two-terminal resp. all-terminal reliability of the network in Figure 6.3 are better than the classical Bonferroni bounds for network reliability, although fewer terms are taken into account. We now compare our abstract tube bounds with some well-known optimal Bonferroni bounds. A good source of information on optimal Bonferroni bounds is Pr´ekopa [Pr´e88, Pr´e90, Pr´e95] where such bounds are obtained by linear programming techniques. See also Galambos and Simonelli [GS96a]. Let (Ω, , P ) be a probability space, and let {Av }v∈V be a ﬁnite family of events with Av ∈ for any v ∈ V . Deﬁne the binomial moments S1 , S2 , . . . by (k = 1, 2, . . . ). Sk := P Ai I⊆V |I|=k

i∈I

The ﬁrst inequality, that we consider, is due to Dawson and Sankoﬀ [DS67]: + * 2 2 2S2 P S1 − S2 where θ := 1 + (6.7) . Av ≥ θ+1 θ(θ + 1) S1 v∈V

The second inequality, which gives an upper bound, is due to Kwerel [Kwe75a]: 2t − 1 t−1 P (6.8) Av ≤ S1 − t+1 S2 + t+1 S3 (t = 2, . . . , |V | − 1). v∈V

2

3

Here, the optimal choice for t is topt := min{2 + 3S3 /S2 , |V | − 1}. Note that the particular case where t = |V | − 1 can also be deduced from our chordal graph sieve (for details, see Section 4.4, p. 42).

6.2 Special reliability systems

65

Let Σ = (E, φ) be a general coherent binary system where each component e ∈ E fails randomly and independently with probability qe = 1 − pe , and let p = (pe )e∈E . The Dawson-Sankoﬀ bound (6.7) gives RelΣ (p) ≥

(6.9)

2 2 S1 − S2 θ+1 θ(θ + 1)

where (6.10)

Sk =

( e∈Ë

* pe

(k = 1, 2, . . . ) ,

θ := 1 +

⊆ | |=k

+ 2S2 , S1

and where denotes the set of minpaths of Σ. In a similar way, by considering mincuts instead of minpaths we obtain from (6.7) RelΣ (p) ≤ 1 −

(6.11)

2 2 S + S θ + 1 1 θ (θ + 1) 2

where (6.12)

Sk

=

( ⊆ e ∈ Ë ||=k

qe

(k = 1, 2, . . . ) ,

+ 2S2 , θ := 1 + S1 *

and denotes the set of mincuts of Σ. Likewise, by Kwerel’s inequality (6.8), (6.13)

2topt − 1 topt − 1 RelΣ (p) ≤ S1 − topt +1 S2 + topt +1 S3 2

3

where S1 , S2 , S3 are given by (6.10), topt = min{2 + 3S3 /S2 , | | − 1} and denotes the set of minpaths of Σ. Similar to the above, (6.8) gives (6.14)

2topt − 1 topt − 1 RelΣ (p) ≥ 1 − S1 + t +1 S2 − t +1 S3 opt

2

opt

3

where S1 , S2 , S3 are given by (6.12), topt = min{2 + 3S3 /S2 , | | − 1} and denotes the set of mincuts of Σ. Now, for the two-terminal reliability of the network in Figure 6.3 we have (6.15)

S1 = 3p2 + 4p3 + 2p4 ,

S1 = 2q 3 + 4q 4 + 2q 5 ,

(6.16)

S2 = 11p4 + 16p5 + 9p6 ,

S2 = 4q 5 + 13q 6 + 10q 7 + q 8 ,

(6.17)

S3 = 8p5 + 43p6 + 30p7 + 3p8 , S3 = 32q 7 + 24q 8 ,

provided that all edges fail with equal probability q = 1 − p. Let aDS and bDS 2 2 be the Dawson-Sankoﬀ bounds given by (6.9) and (6.11), respectively, where Si and Si are taken from (6.15), (6.16) and (6.17), and where we use the subscript 2 in aDS and bDS in order to signify that these bounds are of order 2. Likewise, 2 2

66

6

Reliability Applications

let aKw and bKw be the Kwerel bounds given by (6.13) and (6.14), respectively, 3 3 where use the subscript 3 in order to signify that these bounds are of order 3. For the all-terminal reliability of the network in Figure 6.3 we ﬁnd that (6.18)

S1 = 4q 3 + 5q 4 + 4q 5 ,

(6.19)

S2 = 12q 5 + 44q 6 + 20q 7 + 2q 8 ,

(6.20)

S3 = 4q 5 + 14q 6 + 180q 7 + 88q 8 ,

provided that all edges fail with equal probability q = 1 − p. Let dDS be the 2 Dawson-Sankoﬀ bound given by (6.11) and dKw be the Kwerel bound given 3 by (6.14), where this time S1 , S2 , S3 are taken from (6.18), (6.19) and (6.20), respectively. Again, subscripts are used to indicate the order of the bounds. In Table 6.7 some numerical values are given that allow us to compare our abstract tube bounds with Dawson-Sankoﬀ and Kwerel bounds for the speciﬁc network in Figure 6.3. Note that the second and the third column in Table 6.7 provide lower bounds to the reliability of the network whereas the ﬁfth and the sixth column provide upper bounds. It turns out that for small values of p resp. q the abstract tube bounds are at least as sharp as the DawsonSankoﬀ and Kwerel bounds, whereas for large values of p resp. q the DawsonSankoﬀ and Kwerel bounds are sharper than our abstract tube bounds. In most practical applications, p is large and q is small. Thus, it seems that the abstract tube bounds are advantageous if cutsets are used to estimate the reliability of the network, while the Dawson-Sankoﬀ and Kwerel bounds are advantageous if pathsets are used instead. Note, however, that in both cases the abstract tube bounds require much fewer sets to be inspected during their computation.

6.2.2

k-out-of-n systems

The class of k-out-of-n systems was introduced by Birnbaum, Esary and Saunders [BES61]; see also Rushdi [Rus93] for an extensive survey. A k-out-of-n success (resp. failure) system operates (resp. fails) whenever k or more components operate (resp. fail). As for coherent systems, it is again assumed that the components fail randomly and independently (this assumption might be relaxed). A formal deﬁnition in terms of coherent binary systems follows. Definition 6.2.2 Let k, n ∈ and 1 ≤ k ≤ n. A k-out-of-n success (resp. failure) system is a coherent binary system Σ = (E, φ) where |E| = n and where for any subset X of E, φ(X) = 1 (resp. φ(E \ X) = 0) if and only if |X| ≥ k. Remark. The minpaths (resp. mincuts) of a k-out-of-n success (resp. failure) system Σ = (E, φ) are the k-subsets of E. Likewise, the mincuts (resp. minpaths) of a k-out-of-n success (resp. failure) system Σ = (E, φ) are the (n − k + 1)-subsets of E. Any k-out-of-n success (resp. failure) system is an (n − k + 1)-out-of-n failure (resp. success) system. Thus, we may restrict ourselves to either type of system.

6.2 Special reliability systems

p

a∗2

aDS 2

Rel†

aKw 3

a∗3

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.033 0.133 0.270 0.354 0.219 -0.404 -1.907

0.033 0.132 0.260 0.364 0.474 0.587 0.690

0.033 0.136 0.297 0.487 0.672 0.823 0.925

0.033 0.138 0.311 0.540 0.796 1.053 1.268

0.033 0.136 0.301 0.520 0.829 1.372 2.506

† two-terminal

reliability

q

b∗3

bKw 3

Rel†

bDS 2

b∗2

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.998 0.979 0.925 0.819 0.648 0.386 -0.049

0.998 0.979 0.922 0.808 0.621 0.363 0.063

0.998 0.979 0.925 0.823 0.672 0.487 0.297

0.998 0.979 0.930 0.860 0.762 0.649 0.515

0.998 0.979 0.928 0.847 0.781 0.868 1.384

† two-terminal

reliability

q

d∗3

dKw 3

Rel†

dDS 2

d∗2

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.996 0.963 0.874 0.708 0.430 -0.089 -1.241

0.996 0.962 0.868 0.687 0.408 0.045 -0.343

0.996 0.963 0.875 0.723 0.523 0.314 0.142

0.996 0.966 0.908 0.829 0.736 0.629 0.499

0.996 0.963 0.885 0.794 0.852 1.456 3.403

† all-terminal

67

reliability

Table 6.7: Abstract tube bounds vs. Dawson-Sankoﬀ-Kwerel bounds.

As in [Doh98] we establish a partial ordering relation on the set of k-subsets of E, thereby assuming that E is endowed with a linear ordering relation ≤ . For k-subsets X and Y of a linearly ordered set E deﬁne (6.21)

X≤Y

:⇔

x ≤ y for all x ∈ X, y ∈ Y \ X .

Thus, a partial ordering relation on the set of k-subsets of E is established. Figure 6.10 shows the associated Hasse diagram for E = {1, . . . , 6} and k = 3. Again, it is easy to see that the convexity requirement of Theorem 6.1.6 is satisﬁed; moreover, X ∧ Y ⊆ X ∪ Y , since X ∧ Y consists of the k smallest elements of X ∪Y . Therefore, the results of Section 6.1 can be applied to k-out-of-n success or failure systems whose k-subsets are ordered as in (6.21). We conclude that for ﬁxed k, the time and space complexity of the pseudopolynomial algorithm (Algorithm II), when applied to the k-subsets of an n-set, are O(n2k+1 ),

68

6

126

Reliability Applications

136

146

156

236

246

256

346

125

135

145

235

245

345

124

134

356

456

234

123

Figure 6.10: Hasse diagram for E = {1, . . . , 6} and k = 3.

resp. O(nk ). A more eﬃcient method is the generating function method of Barlow and Heidtmann [BH84], whose time complexity is linear in n if k is ﬁxed. For k-out-of-n success or failure systems Σ = (E, φ) we subsequently consider the number of terms in the improved inclusion-exclusion identity associated with the abstract tube of Theorem 6.1.3, that is, we compute the number of nonempty chains in the partially ordered set of k-subsets of E. Theorem 6.2.3 [Doh98] Let E be a finite set of cardinality n, whose k-subsets are ordered as in (6.21). Then, the number of non-empty chains in the poset of k-subsets of E is equal to 2fk (n − k) − 1, where fk (0) := 1 and (6.22)

t−1 t−s+k−1 fk (s) fk (t) := 1 + k−1 s=0

(t = 1, . . . , n − k) .

Proof. For any k-subset I of E, let c(I) denote the number of chains extending upward from I. Then, the total number of chains is 2c(ˆ0) where ˆ0 = {1, . . . , k} denotes the minimum of the poset. We show that c(ˆ0) = fk (n − k). More generally, by induction on t we prove h(I) = n − k − t ⇒ c(I) = fk (t),

(6.23)

where h(I) denotes the height of I in the poset of k-subsets of E. For t = 0 this is obvious, since n − k is the maximum height of the poset. Now assume that h(I) = n − k − t for some t > 0. Then, by the induction hypothesis, c(I) = 1 +

t−1

s=0

J≥I h(J)=n−k−s

c(J) = 1 +

t−1

s=0

J≥I h(J)=n−k−s

fk (s) = 1 +

t−1 s=0

Ns (I)fk (s)

6.2 Special reliability systems

69

where Ns(I) := |{J ≥ I | h(J) = n − k − s}| (s = 0, . . . , t). It is easy to see that Ns (I) = t−s+k−1 (hint: use downward induction on s), and therefore, k−1 t−1 t−s+k−1 fk (s) = fk (t) , c(I) = 1 + k−1 s=0 which settles (6.23). Thus, the proof is complete. 2 In order to compare the number of terms in the improved inclusion-exclusion identity with the number of terms in the classical inclusion-exclusion expansion for ﬁxed k and increasing n, it seems reasonable to consider the ratio

k (n) :=

(6.24)

2fk (n − k) − 1 , n 2( k ) − 1

with fk (t) as in (6.22). It should be clear, though, that the classical inclusionexclusion method is not an eﬀective method for analyzing k-out-of-n systems. The following numerical values are computed via (6.22) and (6.24):

2 (6) ≈ 7.0 × 10−3 ,

3 (6) ≈ 1.7 × 10−4 ,

4 (6) ≈ 1.8 × 10−3 ,

2 (8) ≈ 1.0 × 10−5 ,

3 (8) ≈ 6.0 × 10−14 ,

4 (8) ≈ 2.1 × 10−18 ,

2 (10) ≈ 9.0 × 10−10 ,

3 (10) ≈ 7.6 × 10−32 ,

4 (10) ≈ 5.9 × 10−59 .

This suggests that k (n) tends to 0 as n → ∞. In fact, this is the statement of the following theorem. Theorem 6.2.4 [Doh98] For any k > 1, limn→∞ k (n) = 0. Proof. By Theorem 6.2.3 and since fk (t) ≤ 1 +

t−1

t−s+k−1 k−1

k t−s fk (s)

≤ k t−s we immediately ﬁnd that (t = 0, . . . , n − k) ,

s=0

and hence, fk (t) ≤ 1 + k

t−1

(2k)s = 1 + k

s=0

1 − (2k)t 1 − 2k

(t = 0, . . . , n − k) .

Hence, for ﬁxed k, there are constants c and d depending only on k such that

k (n) ≤ c

k (2k)n ∼ c2dn−n , n ) ( k 2

which immediately implies the statement of the theorem. 2

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Remark. An explicit solution for fk (t) for general ∞k is not known. However, we can compute the generating function Fk (z) := t=0 fk (t)xt . By (6.22), ∞ t−1 t−s+k−1 1+ fk (s) z t Fk (z) = k − 1 t=0 s=0 ∞ t−1 t−s+k−1 1 + fk (s) z t = 1 − z t=0 s=0 k−1 t ∞ t−s+k−1 1 + fk (s) z t − Fk (z), = 1 − z t=0 s=0 k−1 whence

∞ 1 t+k−1 t + Fk (z) z 1−z k−1 t=0 1 1 1 + Fk (z) . = 2 1−z (1 − z)k

1 Fk (z) = 2

Solving for Fk (z) we obtain Fk (z) =

(1 − z)k−1 . 2(1 − z)k − 1

For instance, for k = 3 we obtain the series expansion F3 (z) = 1 + 4z + 19z 2 + 92z 3 + 446z 4 + 2162z 5 + 10480z 6 + 50800z 7 + · · · From f3 (3) = 92 we conclude that the semilattice depicted in Figure 6.10 contains 2f3 (3) − 1 = 183 non-empty chains. So we again have 3 (6) ≈ 1.7 × 10−4 . Note. The derivation of the generating function is due to Peter Tittmann (Mittweida), whom I would like to thank very much for his worthy contribution.

6.2.3

Consecutive k-out-of-n systems

Consecutive k-out-of-n success (resp. failure) systems operate (resp. fail) whenever k or more consecutive components operate (resp. fail). Again, it is assumed that the components fail randomly and independently with known probabilities. Systems of this type were ﬁrst considered by Kontoleon [Kon80]; the nomenclature goes back to Chiang and Niu [CN81], who also provide several applications of this model. For an account of consecutive systems, we recommend the survey paper of Papastavridis and Koutras [PK93]. A formal deﬁnition follows. Definition 6.2.5 Let k, n ∈ , 1 ≤ k ≤ n. A consecutive k-out-of-n success (resp. failure) system is a coherent binary system Σ = (E, φ) where E is a linearly ordered ﬁnite set of size n and where for any subset X of E, φ(X) = 1 (resp. φ(E \ X) = 0) if and only if X contains at least k consecutive elements of E.

6.2 Special reliability systems

71

As illustrated by the following example, consecutive k-out-of-n failure systems serve as a model for a particular type of communications networks. Example 6.2.6 In the network of Figure 6.11 the nodes 1–7 are assumed to fail randomly and independently with known probabilities whereas the edges and the two terminal nodes s and t are perfectly reliable. Evidently, in this network a message can pass from s to t unless four consecutive nodes among 1–7 simultaneously fail. Thus, we are faced with a 4-out-of-7 failure system.

s

1

2

3

4

5

6

7

t

Figure 6.11: A consecutive 4-out-of-7 failure network. In general, X is a minpath (resp. mincut) of a consecutive k-out-of-n success (resp. failure) system Σ = (E, φ) if and only if X is a consecutive subset of E containing exactly k elements. The following partial ordering relation on the set of consecutive k-subsets of E is adopted from Shier [Shi88, Shi91]: For any consecutive k-subsets X and Y of a linearly ordered set E deﬁne (6.25)

X≤Y

:⇔

min X ≤ min Y .

Thus, a partial (in fact: linear) ordering relation on the set of minpaths (resp. mincuts) of a consecutive k-out-of-n success (resp. failure) system is given, which satisﬁes the requirements of Theorem 6.1.6: If X and Y are two consecutive ksubsets of E, then X ∧ Y = X or X ∧ Y = Y and hence, X ∧ Y ⊆ X ∪ Y . If X ≤ Y ≤ Z are three consecutive k-subsets of E and e ∈ X ∩ Z, then min Y ≤ min Z ≤ e ≤ max X ≤ max Y and hence e ∈ Y . Therefore, the requirements of Theorem 6.1.6 are satisﬁed and thus Algorithm II can be applied. Since the number of minpaths (resp. mincuts) is n − k + 1, the algorithm has a space complexity of O(n) and a time complexity of O(n3 ). In contrast, the time complexity of the classical inclusion-exclusion method for that problem is O(n2n ). Note, however, that the classical method can be generalized to dependent component failures, whereas Algorithm II strongly relies on the independence assumption. Similar remarks as for the classical inclusionexclusion method applies to the inclusion-exclusion expansion of Kossow and Preuss [KP89], which involves only O(n4 ) non-cancelling terms, and which is best possible among all inclusion-exclusion expansions for the reliability of a consecutive k-out-of-n system with unequal component reliabilities. For k ≥ n/2 the following theorem provides an improved inclusion-exclusion expansion that contains only O(n2 ) terms none of which cancel. Thus, the expansion is best possible among all inclusion-exclusion expansions for this restricted case.

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Theorem 6.2.7 [Doh03] Let Σ be a consecutive k-out-of-n success system whose component reliabilities are given by the vector p = (p1 , . . . , pn ). If k ≥ n/2, then (6.26)

RelΣ (p) =

n−k+1 i+k−1 ( i=1

pj −

j=i

n−k i+k (

pj .

i=1 j=i

Proof. For i = 1, . . . , n−k +1 let Ai be the event that components i, . . . , i+k −1 operate. Then, RelΣ (p) = Pr(A1 ∪ · · · ∪ An−k+1 ). Since k ≥ n/2, Ax ∩ Ay ⊆ Az for x, y = 1, . . . , n − k + 1 and any z between x and y. By Corollary 4.1.12, RelΣ (p) =

n−k+1

P (Ai ) −

i=1

where P (Ai ) =

)i+k−1 j=i

n−k

P (Ai ∩ Ai+1 ) ,

i=1

pj and P (Ai ∩ Ai+1 ) =

)i+k j=i

pj . 2

Remark. Note that (6.26) can also be written as (6.27)

RelΣ (p) =

n−k i=1

(1 − pi+k )

i+k−1 ( j=i

pj +

n (

pj ,

j=n−k+1

and that both identities can be generalized to dependent component failures. For independent components, the expression in (6.27) can be computed in O(n) steps by means of the following algorithm. Algorithm III Reliability of a consecutive k-out-of-n system where k ≥ n/2 Require: Components 1, . . . , n work independently with probability p1 , . . . , pn Ensure: acc gives the probability that at least k consecutive components work 1: acc ← 0 2: h ← p1 . . . pk 3: for i = 1 to n − k do 4: acc ← acc + (1 − pi+k )h 5: h ← hpi+k /pi 6: end for 7: acc ← acc + h In the case of equal component reliabilities we even obtain a closed formula: Corollary 6.2.8 [Doh03] Let Σ = (E, φ) be a consecutive k-out-of-n success system with component reliabilities given by p = (p, . . . , p). If k ≥ n/2, then RelΣ (p) = (n − k + 1)pk − (n − k)pk+1 = pk [(n − k)(1 − p) + 1] . Proof. Corollary 6.2.8 is an immediate consequence of Theorem 6.2.7. 2

6.2 Special reliability systems

73

Remarks. Theorem 6.2.7 and Corollary 6.2.8 can equivalently be formulated for consecutive k-out-of-n failure systems by replacing the preceding identities with n−k+1 i+k−1 (

1 − RelΣ (p) =

i=1

j=i

qj −

n−k i+k ( i=1 j=i

qj =

n−k

(1 − qi+k )

i=1

i+k−1 (

qj +

j=i

n (

qj

j=n−k+1

in the case of unequal component failure probabilities qj = 1 − pj , and 1 − RelΣ (p) = (n − k + 1)q k − (n − k)q k+1 = q k [(n − k)(1 − q) + 1] in the case where all component failure probabilities are equal to q = 1 − p. This latter identity was ﬁrst proved by Shanthikumar [Sha82] without making use of the inclusion-exclusion principle. It should be noted at this point that the algorithms of Shanthikumar [Sha82] and Hwang [Hwa82] are the most eﬃcient algorithms for computing the reliability of a consecutive k-out-of-n system. Note, however, that these most eﬃcient algorithms strongly rely on the assumption that the components of the system fail in a statistically independent fashion.

Example 6.2.9 As in Example 6.2.6 we consider the consecutive 4-out-of-7 failure system associated with the network in Figure 6.11. In view of the preceding remarks the reliability of this consecutive 4-out-of-7 failure system is 1−q1 q2 q3 q4 −q2 q3 q4 q5 −q3 q4 q5 q6 −q4 q5 q6 q7 +q1 q2 q3 q4 q5 +q2 q3 q4 q5 q6 +q3 q4 q5 q6 q7 = 1 − (1 − q5 )q1 q2 q3 q4 − (1 − q6 )q2 q3 q4 q5 − (1 − q7 )q3 q4 q5 q6 − q4 q5 q6 q7 , where qi denotes the failure probability of node i (i = 1, . . . , 7). In particular, if the qi ’s are all equal to q, then the reliability is easily seen to be 1 − 4q 4 + 3q 5 . We close this section with a generalization of Theorem 6.2.7, to which the remarks after Corollary 6.2.8 likewise apply. Theorem 6.2.10 Let t ∈ and n1 , . . . , nt , k1 , . . . , kt ∈ such that n1 < · · · < nt ≤ n1 +k1 < · · · < nt +kt . Let Σ be a coherent binary system with components n1 , . . . , nt + kt − 1, minpaths {n1 , . . . , n1 + k1 − 1}, . . . , {nt , . . . , nt + kt − 1} and component reliabilities given by p = (pn1 , . . . , pnt +kt −1 ). Then, RelΣ (p) equals   ni+1 +ki+1 −1 ni +k t ni +k t−1 ni+1 +k t−1 (i −1 (i+1 −1 ( (i −1 nt +k (t −1 1 − pj − pj = pj  pj + pj . i=1

j=ni

i=1

j=ni

i=1

j=ni +ki

j=ni

j=nt

Proof. For i = 1, . . . , t let Ai be the event that components ni , . . . , ni + ki − 1 operate. It follows that RelΣ (p) = P (A1 ∪ · · · ∪ At ) and, by the requirements of the theorem, Ax ∩ Ay ⊆ Az for x, y = 1, . . . , t and any z between x and y.

74

6

Reliability Applications

Therefore, the same argument as in the proof of Theorem 6.2.7 reveals that RelΣ (p) =

t

P (Ai ) −

i=1

=

=

P (Ai ∩ Ai+1 ) =

i=1 t−1 ni+1 +k (i+1 −1

i=1

i=1

pj −

j=ni

t−1 ni+1 +k (i+1 −1 pj − pj

t ni +k (i −1 i=1

t−1 ni +k (i −1

pj +

j=ni

i=1

j=ni

nt +k (t −1

j=ni

pj

j=nt

nt +k t−1 ni +k t−1 ni +k (i −1 (i −1 ni+1 +k (i+1 −1 (t −1 pj − pj pj + pj i=1

=

t−1

j=ni

t−1



1 −

i=1 ni+1 +ki+1 −1

(



pj 

j=ni +ki

i=1

j=ni +ki

j=ni

j=nt

ni +k (i −1

nt +k (t −1

j=ni

j=nt

pj +

pj .

2

Example 6.2.11 Consider the network in Figure 6.12, where nodes 1–7 are assumed to fail randomly and independently with probabilities q1 , . . . , q7 and all other nodes and edges are perfectly reliable. Again, we are interested in the probability that a message can pass from s to t along a path of operating nodes. Thus, an appropriate model is a coherent binary system having components 1– 7 and mincuts {1, 2, 3}, {2, 3, 4}, {3, 4, 5, 6}, {4, 5, 6, 7}. By applying the mincut analogue of Theorem 6.2.10 the reliability of this system is easily seen to be 1 − q1 q2 q3 − q2 q3 q4 − q3 q4 q5 q6 − q4 q5 q6 q7 + q1 q2 q3 q4 + q2 q3 q4 q5 q6 + q3 q4 q5 q6 q7 = 1 − (1 − q4 )q1 q2 q3 − (1 − q5 q6 )q2 q3 q4 − (1 − q7 )q3 q4 q5 q6 − q4 q5 q6 q7 , which equals 1 − 2q 3 − q 4 + 2q 5 if all node failure probabilities are equal to q.

s

1

2

3

4

5

6

7

t

Figure 6.12: A consecutive network.

Remark. The inclusion-exclusion identity of Theorem 6.2.10 contains only noncancelling terms. The coeﬃcient of each such term is either +1 or −1. Due to Shier and McIlwain [SM91], this ±1 property holds for any coherent binary system whose minpaths (resp. mincuts) are consecutive sets of components. This strongly generalizes a corresponding result of Kossow and Preuss [KP89], who showed that this ±1 property holds for any consecutive k-out-of-n system.

6.3

6.3

Reliability covering problems

75

Reliability covering problems

Reliability covering problems were introduced by Ball, Provan and Shier [BPS91, Shi91] as an abstract model for several types of reliability problems such as evaluating the probability that in a mass transit system with reliable stops and unreliable routes each stop is served by a route. Further examples include determining the reliability of ﬂight schedules for aircraft [BPS91] and of maintaining continuous surveillance of a critical segment of a country’s border [Shi91].

6.3.1

The hypergraph model

As in [BPS91, Shi91] we use the terminology of hypergraphs: Definition 6.3.1 A hypergraph is a couple H = (V, ) where V is a ﬁnite set and is a set of non-empty subsets of V . The elements of V resp.

are the vertices resp. edges of H. A covering of V is a subset of such that = V . In case of a mass transit system, the vertices and edges of the hypergraph correspond to the stops and routes of the system, respectively, while the coverings correspond to those sets of routes such that each stop is served by a route. Throughout, we make the assumption that the vertices of H are perfectly reliable, whereas the edges fail randomly and independently with known probabilities, which are given by a vector p = (pE )E∈ ∈ [0, 1] . The general objective is to determine or compute bounds on the probability that the operating edges of H cover the whole vertex-set of H, which is abbreviated to Cov(H; p). Besides their practical applicability, reliability covering problems are also interesting from a theoretical point of view. Namely, any coherent binary system gives rise to an equivalent reliability covering problem, and vice versa [BPS91, Shi91]. In this way, the results of the preceding sections (including those on network reliability) can be reformulated as reliability covering problems. One direction of this equivalence is used in deriving the following results on reduced hypergraphs, i.e., hypergraphs of type H = (V, ) where the sets (v) := {E ∈ | v ∈ E} (v ∈ V ) are all distinct. Note that restricting to reduced hypergraphs does not cause any loss of generality, since by deleting vertices any hypergraph H can be transformed eﬃciently into a reduced hypergraph R(H) such that Cov(H; p) = Cov(R(H); p). Now, using the above sets, the coverage probability can be expressed as   (6.28) Cov(H; p) = 1 − P  {E fails} . v∈V E∈(v)

6.3.2

Abstract tubes and polynomial algorithms

In connection with Theorem 3.1.9 the ﬁrst part of the following theorem, which is implicit in [Doh99b], yields improved inclusion-exclusion identities and Bonferroni inequalities for the last term in (6.28) and thus for the coverage probability

76

6

Reliability Applications

Cov(H; p). We do not mention the improved inclusion-exclusion identities explicitly, since they follow immediately from the corresponding inequalities. As in the preceding sections, we use (V ) to denote the order complex of V . Theorem 6.3.2 [Doh99b] Let H = (V, ) be a reduced hypergraph whose edges fail randomly and independently and whose vertex-set is given the structure of a lower semilattice such that the complement of each edge is infimum-closed. Then, , {E fails} , (V ) E∈(v)

v∈V

is an abstract tube. In particular, for any p = (pE )E∈ ∈ [0, 1] and any r ∈ , (6.29)

Cov(H; p) ≤

(−1)|I|

Cov(H; p) ≥

qE

(r even),

qE

(r odd),

E∈ E∩I=∅

I⊆V,|I|≤r I is a chain

(6.30)

(

(−1)|I|

I⊆V,|I|≤r I is a chain

(

E∈ E∩I=∅

where qE = 1 − pE for any E ∈ . Proof. Consider the coherent binary system Σ = (, φ) where for any subset of , φ( \ ) = 0 if and only if ⊇ (v) for some v ∈ V . Clearly := {(v) | v ∈ V } is an extended set of mincuts of this system. Now, by deﬁning (v) ≤ (w) :⇔ v ≤ w for any v, w ∈ V , a partial ordering relation on is established, which satisﬁes (v) ∧ (w) = (v ∧ w) for any v, w ∈ V . Since the complement of each edge is inﬁmum-closed, (v ∧ w) ⊆ (v) ∪ (w) and therefore, (v) ∧ (w) ⊆ (v) ∪ (w) for any v, w ∈ V . Hence, the requirements of the extended mincut version of Theorem 6.1.3 are satisﬁed, and thus the ﬁrst part of the theorem follows. The second part follows from Theorem 3.1.9. 2 Remarks. Theorem 6.3.2 can be generalized to reduced hypergraphs with statistically dependent edge failures: Simply replace the product in (6.29) and (6.30) with the probability that all edges having a non-empty intersection with I fail. Note that the requirements of Theorem 6.3.2 (as well as those of the next theorem) are satisﬁed if H = ({1, . . . , n}, ) where ⊆ {{k, . . . , l} | 1 ≤ k ≤ l ≤ n} and n ∈ . For an application of this type of hypergraph, see Shier [Shi91]. Theorem 6.3.3 [BPS91, Shi91] Let H = (V, ) be a reduced hypergraph whose edges fail randomly and independently according to some vector q = (qE )E∈ of edge failure probabilities and whose vertex-set is a lower semilattice such that each edge is convex and the complement of each edge is infimum-closed. Then, Cov(H; p) = 1 − (6.31) Λ(v, q) , v∈V

6.3

Reliability covering problems

77

where p = 1 − q and where Λ is defined by the following recursive scheme: (6.32)

Λ(v, q) :=

( E∈(v)

qE −

Λ(w, q)

(

qE .

E∈ (v) E∈ / (w)

w
Proof. Theorem 6.3.3 follows from Theorem 6.1.6 in the same way as Theorem 6.3.2 follows from Theorem 6.1.3. 2 Remark. In view of equations (6.31) and (6.32), Algorithm II can be adapted to solve the reliability covering problem for the class of hypergraphs H = (V, ) satisfying the requirements of Theorem 6.3.3. The resulting algorithm, which is due to Ball, Provan and Shier [BPS91, Shi91], has a space complexity of O(|V |) and a time complexity of O(|| × |V |2 ), that is, its running time is bounded by a polynomial in the number of vertices and edges of the hypergraph. Thus, under the requirements of Theorem 6.3.3, the reliability covering problem can be solved in polynomial time. In general, though, the reliability covering problem is #P -hard, even when restricted to hypergraphs whose vertices are the vertices of a tree and whose edges are paths of length three in the tree [BPS91, Shi91]. However, if the vertices are the elements of a lower tree semilattice (that is, a lower

semilattice whose Hasse diagram is a tree) and the edges are of the form k {v ∈ V | xk ≤ v, yk ≤ v} where xk ≤ yk and the xk ’s are pairwise incomparable, then the requirements of Theorem 6.3.3 are satisﬁed and the reliability covering problem can be solved in polynomial time [BPS91, Shi91]. We proceed with establishing a generalization of Theorem 6.3.2. Theorem 6.3.4 [Doh03] Let (V, c) be a convex geometry, and let H = (V, ) be a hypergraph on V with randomly failing edges such that the complement of each edge is c-closed. Then, , {E fails} , Free(V, c) E∈(v)

v∈V

is an abstract tube. In particular, if the edges fail in an independent manner, then for any p = (pE )E∈ ∈ [0, 1] and r ∈ we have the inequalities ( Cov(H; p) ≤ (−1)|I| qE (r even),

E∈ E∩I=∅

I⊆V,|I|≤r I is c-free

Cov(H; p) ≥

I⊆V,|I|≤r I is c-free

(

(−1)|I|

qE

(r odd),

E∈ E∩I=∅

where, as above, qE = 1 − pE for any E ∈ . Proof. For any v ∈ V let Av be the event that v becomes isolated. For any X ∈ ∗ (V ) and any v ∈ c(X) we show that x∈X Ax ⊆ Av , thus establishing

78

6

Reliability Applications

the requirements of Theorem 4.1.9, which proves the ﬁrst part of the theorem. Obviously, x∈X Ax ⊆ Av if E ∩ X = ∅ for all E ∈ (v). This latter condition is now proved by contradiction: Assume that E ∩ X = ∅ for some E ∈ (v). Then, X ⊆ E, and since by the requirements of the theorem E is c-closed, we also ﬁnd that v ∈ c(X) ⊆ E, which is in contradiction with E ∈ (v) (i.e., v ∈ E). Thus, the ﬁrst part of Theorem 6.3.4 is proved. The second part of Theorem 6.3.4 now follows from its ﬁrst part, Theorem 3.1.9 and (6.28). 2 As already mentioned above, the problem of computing the coverage probability Cov(H; p) of any hypergraph is #P -hard, even when the vertices of the hypergraph are the vertices of a tree and its edges are paths of length three in the tree [BPS91, Shi91]. In fact, it does not matter whether the tree is part of the input or not. As the following theorem shows, the situation becomes totally diﬀerent if we consider complements of paths or subtrees instead of paths. Theorem 6.3.5 [Doh03] The coverage probability of any hypergraph whose vertices are the vertices of a tree and whose edges are complements of subtrees of the tree can be computed in polynomial time from the hypergraph and the tree. Proof. Let H = (V, ) be a hypergraph such that each E ∈ is the complement of a subtree of some tree G = (V, T ). By combining Theorem 6.3.4 with Example 4.1.6 (or by applying Corollary 4.1.12) we obtain the formula ( ( (6.33) Cov(H; p) = 1 − qE + qE , v∈V E∈(v)

{v,w}∈T E∈(v)∪(w)

which can be evaluated in O(|V | · ||) steps. 2 An even more general result is the following. Theorem 6.3.6 [Doh03] The coverage probability of any hypergraph whose vertices are those of a connected block graph of bounded clique number and whose edges are complements of connected subgraphs of the connected block graph can be computed in polynomial time from the hypergraph and the connected block graph. Proof. Let c ∈ , and let H = (V, ) be a hypergraph such that each edge E ∈ is the complement of a connected subgraph of some connected block graph G whose clique number is bounded by c. By applying Theorem 6.3.4 in connection with the convex geometry of Example 4.1.7 we obtain the following formula: ( (6.34) Cov(H; p) = (−1)|I| qE . I is a clique of G

E∈ E∩I=∅

Evidently, this formula can be evaluated in O(|V |c · ||) steps. 2 The next example shows an application of the improved Bonferroni inequalities of Theorem 6.3.4 in connection with the convex geometry of Example 4.1.7.

6.3

Reliability covering problems

79

Example 6.3.7 Let H be the hypergraph whose vertex-set is given by {a, b, c, d, e, f, g}, and whose edge-set is {{a, b, c, d}, {d, e, f, g}, {a, f, g}, {a, c, f }, {b, c, e, g}, {b, e, f }, {b, f, g}, {a, e, g}}. We assume that the edges fail randomly and independently with equal probability q = 1 − p. Evidently, each complement of an edge in this hypergraph determines a connected subgraph of the graph in Figure 6.13. Since this graph is a connected block graph, we can apply Theorem 6.3.4 in connection with Example 4.1.7. The results are shown in Tables 6.8–6.10 and Figure 6.14. Here, er resp. e∗r denote the rth classical resp. improved Bonferroni bound, where even and odd values of r give upper and lower bounds, respectively. As in Example 6.2.1, the improved Bonferroni bounds are much sharper than the classical Bonferroni bounds, although much fewer terms (sets) need to be evaluated. As in Subsection 6.2.1 (see pp. 64–67) we can compute Dawson-Sankoﬀ and Kwerel bounds for the coverage probability of the above hypergraph and compare these bounds with the abstract tube bounds of Table 6.8. The result, which is computed using the binomial moments (see (6.12)) S1 = q 2 + q 3 + 3q 4 + 2q 5 , S2 = q 4 + 5q 5 + 5q 6 + 10q 7 , S3 = 4q 6 + 15q 7 + 16q 8 , is shown in Table 6.11, where eDS 2 comes from the Dawson-Sankoﬀ bound (6.11) and eKw from the Kwerel bound (6.14). Note that the second and third column 3 in Table 6.11 provide lower bounds to the coverage probability of the hypergraph, while the ﬁfth and the sixth column provide upper bounds. Again it turns out that the abstract tube bounds are at least as sharp as the DawsonSankoﬀ and Kwerel bounds if the failure probabilities are small (the more typical case), whereas the Dawson-Sankoﬀ and Kwerel bounds are sharper than the abstract tube bounds for large failure probabilities (the less typical case). a

f

c

b

d

e

g

Figure 6.13: A connected block graph.

80

6

Reliability Applications

classical bounds er

r q2

1− 1 − q2 1 − q2 1 − q2 1 − q2 1 − q2 1 − q2

1 2 3 4 5 6 7

q3

3q 4

sets

2q 5

− − − − q 3 − 2q 4 + 3q 5 + 5q 6 + 10q 7 − q 3 − 2q 4 + 3q 5 + q 6 − 5q 7 − 16q 8 − q 3 − 2q 4 + 3q 5 + q 6 + 14q 8 − q 3 − 2q 4 + 3q 5 + q 6 − 7q 8 − q 3 − 2q 4 + 3q 5 + q 6 − q 3 − 2q 4 + 3q 5 + q 6 − q8

8 29 64 99 120 127 128

Table 6.8: Classical Bonferroni bounds for coverage probability.

improved bounds e∗r

r 1 2 3 4

q2

q3

3q 4

sets

2q 5

1− − − − 1 − q 2 − q 3 − 2q 4 + 3q 5 + 3q 6 + 3q 7 1 − q 2 − q 3 − 2q 4 + 3q 5 + q 6 − 3q 8 1 − q 2 − q 3 − 2q 4 + 3q 5 + q 6 − q8

8 20 28 30

Table 6.9: Improved Bonferroni bounds for coverage probability.

1 Cov 0.8

e2 e*2

0.6

e3 0.4

e*3

0.2

0.2

0.4

0.6

0.8

1

q

Figure 6.14: A plot of some of the bounds in Tables 6.8 and 6.9.

6.4 Chapter notes

q

e3

e∗3

e∗4 †

e4

e∗2

e2

0.1 0.2 0.3 0.4 0.5

0.989 0.950 0.873 0.741 0.508

0.989 0.950 0.875 0.758 0.598

0.989 0.950 0.875 0.759 0.606

0.989 0.950 0.876 0.769 0.664

0.989 0.950 0.877 0.773 0.664

0.989 0.950 0.880 0.793 0.750

† coverage

81

probability

Table 6.10: Numerical values of the bounds in Tables 6.8 and 6.9. q

e∗3

eKw 3

Cov†

eDS 2

e∗2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.989 0.950 0.875 0.758 0.598 0.394 0.136 -0.229

0.989 0.950 0.873 0.748 0.566 0.339 0.080 -0.138

0.989 0.950 0.875 0.759 0.605 0.428 0.251 0.106

0.989 0.950 0.880 0.792 0.708 0.597 0.471 0.331

0.989 0.950 0.877 0.773 0.664 0.622 0.791 1.427

† coverage

probability

Table 6.11: Abstract tube bounds vs. Dawson-Sankoﬀ-Kwerel bounds.

6.4

Chapter notes

Another approach to the analysis of system reliability using abstract tubes has been considered by Giglio, Naiman and Wynn in [GNW02]. The approach relies on a result due to Naiman and Wynn [NW01] that any simplex or orthant arrangement forms an abstract tube with the nerve of an associated Voronoi decomposition. The construction of these abstract tubes is similar to the construction for Euclidean balls using a Voronoi diagram, although the proof of the abstract tube property is more intricate. Using this result, Giglio, Naiman and Wynn [GNW02] establish new inclusion-exclusion expansions and Bonferroni bounds for system reliability by considering coherent systems from a geometric point of view. Their result applies to general multistate systems and requires no further precondition on the structure of the events except that certain associated points should be in general position, which is easily achievable by means of a small perturbation. The algorithmic determination of the corresponding abstract tube is based on the Gr¨obner basis (a common object in computer algebra) of the failure set. For details, the interested reader is referred to [GNW02].

7 Combinatorial Applications and Related Topics We start this chapter with some combinatorial applications of abstract tube theory, and then establish a wide-ranging generalization of the improved inclusionexclusion identity associated with the abstract tube of Theorem 4.2.3 to a general result on sums over partially ordered sets. This general result on sums has applications not only to the inclusion-exclusion principle, but as we will see, also to matroid polynomials, Euler characteristics and M¨ obius functions.

7.1

Inclusion-exclusion on partition lattices

In this section, we establish an abstract tube generalization of Narushima’s principle of inclusion-exclusion on partition lattices [Nar74, Nar77], which has applications to the enumeration of reduced ﬁnite automata [Nar77]. Definition 7.1.1 Let S be a set. A partition π of S is a set of non-empty subsets of S, which are called the blocks of π, such that each element of S is contained in exactly one of these subsets. For any partitions π and τ of S deﬁne π ≤ τ :⇔ each block of π is included in a block of τ , or equivalently, π ≤ τ :⇔ sπs implies sτ s for any s, s ∈ S, where we write sπs to denote that s and s are contained in some common block b of π. In this way, the set of partitions of S, which is denoted by Π(S), is given the structure of a lattice. This lattice structure can be extended to the cartesian product Π(S) × Π(T ) of the lattices Π(S) and Π(T ) associated with any sets S and T by deﬁning for any (π1 , τ1 ), (π2 , τ2 ) ∈ Π(S) × Π(T ), (π1 , τ1 ) ≤ (π2 , τ2 ) :⇔ π1 ≤ π2 and τ1 ≤ τ2 . K. Dohmen: LNM 1826, pp. 82–99, 2003. c Springer-Verlag Berlin Heidelberg 2003

7.2 Chromatic polynomials and broken circuits

83

For any π ∈ Π(S) we refer to (S, π) as a partitioned set. A homomorphism from a partitioned set (S, π) to a partitioned set (T, τ ) is a mapping f : S → T such that sπs implies f (s)τ f (s ) for any s, s ∈ S. We write f : (S, π) → (T, τ ) to denote that f is a homomorphism from (S, π) to (T, τ ). If (S, π) = (T, τ ) then f called an endomorphism of (S, π). We use the following abbreviations: Hom((S, π), (T, τ )) End(S, π)

:= {f | f : (S, π) → (T, τ )} , := Hom((S, π), (S, π)) .

Our abstract tube generalization of Narushima’s principle of inclusion-exclusion on partition lattices [Nar74, Nar77] follows. The inclusion-exclusion identity associated with this abstract tube is due to Narushima [Nar74, Nar77]. Theorem 7.1.2 [Doh01b] Let S and T be finite sets and L be a subsemilattice of Π(S) × Π(T ). Then, {Hom((S, π), (T, τ ))}(π,τ )∈L , (L) is an abstract tube. Proof. The theorem immediately follows from Corollary 4.1.11 by noting that (cf. [Nar74, Nar77]) for any (π, τ ), (π , τ ) ∈ L, Hom((S, π), (T, τ )) ∩ Hom((S, π ), (T, τ )) ⊆ Hom((S, π ∨ π ), (T, τ ∨ τ )) , Hom((S, π), (T, τ )) ∩ Hom((S, π ), (T, τ )) ⊆ Hom((S, π ∧ π ), (T, τ ∧ τ )) , where ∨ and ∧ stand for the supremum (least upper bound) and inﬁmum (greatest lower bound) in Π(S) resp. Π(T ). 2 Corollary 7.1.3 [Doh01b] Let S be a finite set, and let L be a subsemilattice of Π(S). Then, ({End(S, π)}π∈L , (L)) is an abstract tube. Proof. Since the lattices Π(S) and Π(S) × Π(S) are isomorphic (cf. [Gra98]), the corollary follows from the preceding theorem by putting S = T . 2

7.2

Chromatic polynomials and broken circuits

The chromatic polynomial of a graph is a fundamental concept in enumerative combinatorics and chromatic graph theory. It expresses the number of vertexcolorings of a graph in at most λ colors such that no adjacent vertices receive the same color. The chromatic polynomial was introduced by Birkhoﬀ [Bir12] in connection with the four color problem. Since its computation is #P -complete [Val79, Wel93], lower bounds and upper bounds are of great value. Some authors [Bye98, Laz90] justify the seeking of bounds by the analysis of Wilf’s backtracking algorithm for the graph coloring problem [Wil84, Wil86, BW85]. One of the most important results on the chromatic polynomial is Whitney’s broken circuit theorem [Whi32], which relates the coeﬃcients of the chromatic polynomial to the face count numbers of some abstract simplicial complex.

84

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Combinatorial Applications and Related Topics

Our main result in this section is an abstract tube generalization of Whitney’s broken circuit theorem [Whi32], which via Theorem 3.1.9 gives rise to new truncation inequalities for the chromatic polynomial involving the face count numbers of the associated complex. The result is then extended to a new two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial.

7.2.1

The usual chromatic polynomial

The following deﬁnition is standard in chromatic graph theory. Definition 7.2.1 Let G be a graph and λ ∈ . A λ-coloring of G is a mapping f from the vertex-set of G into {1, . . . , λ}. A λ-coloring f of G is proper if f (v) = f (w) for each edge {v, w} of G. We use PG (λ) to denote the number of proper λ-colorings of G and refer to PG (λ) as the chromatic polynomial of G. Remark. The deﬁnition is justiﬁed by a result of Birkhoﬀ [Bir12], who showed that PG (λ) is a monic polynomial in λ of degree n(G) with integer coeﬃcients. Historically, the deﬁnition of the chromatic polynomial was motivated by the four color problem, which asks whether PG (4) > 0 for any planar graph G. To state the main result of this section, a further deﬁnition is needed. Definition 7.2.2 Let G be a graph whose edge-set is endowed with a linear ordering relation. A broken circuit of G is obtained from the edge-set of a cycle of G by removing its maximum edge. The broken circuit complex of G, abbreviated to BC(G), is the abstract simplicial complex consisting of all nonempty subsets of the edge-set of G that do not include any broken circuit of G as a subset. The deﬁnition of a broken circuit goes back to Whitney [Whi32], while the broken circuit complex is due to Wilf [Wil76] (see also [Bry77, BO81, BZ91]). Example 7.2.3 Consider the graph in Figure 7.1, whose edge-set is linearly ordered according to the labelling of the edges. Obviously, the broken circuits are {1, 2}, {1, 2, 4} and {3, 4}, whence the broken circuit complex is equal to {{1}, {2}, {3}, {4}, {5}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 5}, {4, 5}, {1, 3, 5}, {1, 4, 5}, {2, 3, 5}, {2, 4, 5}}.

We now state the main result of this section, which is an abstract tube generalization of Whitney’s broken circuit theorem [Whi32]. For an alternative proof of the associated inequalities, the reader is referred to [Doh99e].

7.2 Chromatic polynomials and broken circuits

1

85

2 3

4

5

Figure 7.1: A graph with labelled edges.

Theorem 7.2.4 [Doh99d] Let G be a graph whose edge-set is endowed with a linear ordering relation. Then, for any λ ∈ , (7.1) {f : V (G) → {1, . . . , λ} | f (v) = f (w)} {v,w}∈E(G) , BC(G) is an abstract tube, and for any r ∈ (7.2) (7.3)

PG (λ) ≥ PG (λ) ≤

r k=0 r

the following inequalities hold:

(−1)k bk (G) λn(G)−k

(r odd),

(−1)k bk (G) λn(G)−k

(r even),

k=0

where b0 (G) = 1 and bk (G), k > 0, counts the faces of cardinality k (dimension k − 1) in BC(G). Proof. Deﬁne as the set of broken circuits of G, and for any edge e of G deﬁne Ae := {f : V (G) → {1, . . . , λ} | f (v) = f (w)} ,

e = {v, w}.

Then, in the sense of Deﬁnition 4.1.18, (E, ) = BC(G). Now, by applying Corollary 4.1.19 (with E instead of V ) we immediately ﬁnd that (7.1) is an abstract tube. From this and Theorem 3.1.9 we conclude that r n(G) k PG (λ) ≥ λ (7.4) + (−1) Ai (r odd), I∈BC(G) k=1

(7.5)

PG (λ)

≤

|I|=k

i∈I

r n(G) k λ + (−1) Ai I∈BC(G) k=1

|I|=k

(r even).

i∈I

Since the edge-subgraph G[I] is cycle-free for any I ∈ BC(G), it follows that (7.6)

m(G[I]) − n(G[I]) + c(G[I]) = 0

86

7

Combinatorial Applications and Related Topics

and hence, Ai = λn(G)−n(G[I]) λc(G[I]) = λn(G)−m(G[I]) = λn(G)−|I| . i∈I

By putting this into (7.4) and (7.5) and collecting powers of λ, the inequalities (7.2) and (7.3) are proved. 2 As a corollary we deduce Whitney’s broken circuit theorem [Whi32]. Corollary 7.2.5 [Whi32] Let G be a graph whose edge-set is endowed with a linear ordering relation. Then, for any λ ∈ ,

n(G)

(7.7)

PG (λ) =

(−1)k bk (G) λn(G)−k ,

k=0

where bk (G) is defined as above. Proof. Corollary 7.2.5 is an immediate consequence of Theorem 7.2.4. 2 As a further corollary we conclude that in absolute value each coeﬃcient of the chromatic polynomial is at most the sum of its left and right neighbour: Corollary 7.2.6 Let the chromatic polynomial of G be given by (7.7). Then, bk (G) ≤ bk−1 (G) + bk+1 (G)

(k = 1, . . . , n(G) − 1).

Proof. We only consider the case where k is odd. Then, by applying Theorem 7.2.4 with λ = 1 and r = k − 2 resp. r = k + 1 we get k−2

(−1)i bi (G) ≤

i=0

k+1

(−1)i bi (G) ,

i=0

which gives the result. 2

7.2.2

The generalized chromatic polynomial

In the sequel, we consider a new two-variable generalization of the chromatic polynomial and establish a corresponding generalization of Theorem 7.2.4. As shown in [DPT03], this new polynomial simultaneously generalizes the chromatic polynomial, the independence polynomial and the matching polynomial, and is closely related to Stanley’s chromatic symmetric function [Sta99]. Definition 7.2.7 [DPT03] Let G = (V, E) be a graph. For any λ ∈ and µ ∈ {0, . . . , λ} we use PG (λ, µ) to denote the number of all mappings f : V → {1, . . . , λ} such that f (v) = f (w) or f (v) = f (w) > µ for any {v, w} ∈ E.

7.2 Chromatic polynomials and broken circuits

87

Remark. Evidently, PG (λ, 0) = λn(G) and PG (λ, λ) = PG (λ). Moreover, PG (2, 1) gives the number of independent sets in G. Recall that a subset W ⊆ V (G) is independent if the vertex-induced subgraph G[W ] has no edges. Theorem 7.2.8 Let G be a graph whose edge-set is endowed with a linear ordering relation, and let λ ∈ and µ ∈ {0, . . . , λ}. Then {f : V (G) → {1, . . . , λ} | f (v) = f (w) ≤ µ} {v,w}∈E(G) , BC(G) is an abstract tube, and for any r ∈ (7.8)

PG (λ, µ) ≥

r k

the following inequalities hold:

(−1)k bk,l (G) λn(G)−k−l µl

(r odd),

(−1)k bk,l (G) λn(G)−k−l µl

(r even),

k=0 l=0

(7.9)

PG (λ, µ) ≤

r k k=0 l=0

where b0,0 (G) = 1 and bk,l (G), k > 0, counts all faces I of cardinality k (dimension k − 1) in BC(G) such that the edge-subgraph G[I] has l connected components. Proof. Deﬁne as the set of broken circuits of G, and for any edge e of G deﬁne Ae := {f : V (G) → {1, . . . , λ} | f (v) = f (w) ≤ µ} ,

e = {v, w}.

As in the proof of Theorem 7.2.4, the ﬁrst part is an immediate consequence of Corollary 4.1.19. In combination with Theorem 3.1.9 the ﬁrst part gives r (7.10) PG (λ, µ) ≥ λn(G) + (−1)k Ai (r odd), I∈BC(G) k=1

(7.11)

PG (λ, µ)

≤ λn(G)

|I|=k

i∈I

r + (−1)k Ai I∈BC(G) k=1

|I|=k

(r even).

i∈I

Since G[I] is cycle-free for any I ∈ BC(G), we again have (7.6) and hence, Ai = λn(G)−n(G[I]) µc(G[I]) = λn(G)−m(G[I])−c(G[I])µc(G[I]) . i∈I

Now, by putting this expression into (7.10) and (7.11) and taking account of c(G[I]) ≤ m(G[I]) = |I|, the inequalities (7.8) and (7.9) are proved. 2

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Combinatorial Applications and Related Topics

Corollary 7.2.9 [DPT03] Let G be a graph whose edge-set is endowed with a linear ordering relation. Then, for any λ ∈ and any µ ∈ {0, . . . , λ}, k

m(G)

(7.12)

PG (λ, µ) =

(−1)k bk,l (G) λn(G)−k−l µl ,

k=0 l=0

where b0,0 (G) = 1 and bk,l (G), k > 0, counts all faces I of cardinality k (dimension k − 1) in BC(G) such that G[I] has l connected components. Proof. Corollary 7.2.9 is an immediate consequence of Theorem 7.2.8. 2 Remarks. Some remarks on Theorem 7.2.8 and Corollary 7.2.9 follow. 1. The preceding equation (7.12) can equivalently be stated as PG (λ, µ) = λn(G) Q G; −λ−1 , µλ−1 , where Q is deﬁned by Q(G; x, y) =

bk,l (G) xk y l .

k,l

Thus, Q(G; −λ−1 , µλ−1 ) expresses the probability that a λ-coloring of G, which is chosen uniformly at random, is admissible in the sense of Deﬁnition 7.2.7. 2. Statements on the number of independent sets are obtained from the preceding results by putting λ = 2 and µ = 1. The corresponding abstract tube is {W | W ⊆ V (G), e ⊆ W } e∈E(G) , BC(G) . 3. We further remark that non-isomorphic trees on the same number of vertices may have diﬀerent polynomials in λ and µ. This contrasts the situation for the usual chromatic polynomial, which equals λ(λ − 1)n−1 for all trees on n vertices. Consider for instance a path G and a star G , both on four vertices. Then, b0,0 (G) = 1,

b2,2 (G) = 1,

b0,0 (G ) = 1,

b2,2 (G ) = 0,

b1,0 (G) = 0, b1,1 (G) = 3,

b3,0 (G) = 0, b3,1 (G) = 1,

b1,0 (G ) = 0, b1,1 (G ) = 3,

b3,0 (G ) = 0, b3,1 (G ) = 1,

b2,0 (G) = 0, b2,1 (G) = 2,

b3,2 (G) = 0, b3,3 (G) = 0,

b2,0 (G ) = 0, b2,1 (G ) = 3,

b3,2 (G ) = 0, b3,3 (G ) = 0.

Putting these values into (7.12) we obtain PG (λ, µ) = λ4 − 3λ2 µ + 2λµ + µ2 − µ, PG (λ, µ) = λ4 − 3λ2 µ + 3λµ − µ. Evidently, these two polynomials diﬀer unless µ = λ or µ = 0. Consequently, the generalized chromatic polynomial PG (λ, µ) is not an evaluation of the Tutte polynomial [Tut47, Tut54], which is the same for all trees on a given number of vertices. We shall be concerned with the Tutte polynomial in Section 7.4.

7.3 Sums over partially ordered sets

7.3

89

Sums over partially ordered sets

In this section we strongly generalize the improved inclusion-exclusion identity associated with the abstract tube of Theorem 4.2.3. The result is a general theorem on sums over partially ordered sets, which has applications not only to the inclusion-exclusion principle, but as we will see in Section 7.4 and Section 7.5, also to matroid polynomials, Euler characteristics and M¨ obius functions.

7.3.1

A general theorem on sums

The following deﬁnition generalizes the concept of a kernel operator resp. closure operator from power set lattices to arbitrary partially ordered sets. Definition 7.3.1 Let P be a partially ordered set. A mapping k : P → P is a kernel operator if for any x, y ∈ P , (i) (ii) (iii)

k(x) ≤ x (intensionality), x ≤ y ⇒ k(x) ≤ k(y) (monotonicity), k(k(x)) = k(x) (idempotence).

Dually, a mapping c : P → P is a closure operator if for any x, y ∈ P , (i) (ii) (iii)

x ≤ c(x)

(extensionality),

x ≤ y ⇒ c(x) ≤ c(y) (monotonicity), c(c(x)) = c(x) (idempotence).

An element x ∈ P is called k-open if k(x) = x and c-closed if c(x) = x. Kernel and closure operators for partially ordered sets were introduced by Ward [War42] and extensively studied by Rota [Rot64]; see also Aigner [Aig79]. The main result of this section is the following: Theorem 7.3.2 [Doh99f] Let P be an upper-finite partially ordered set, and let f and g be mappings from P into an abelian group such that f (x) = y≥x g(y) for any x ∈ P . Furthermore, let k : P → P be a kernel operator, and let x0 be a k-open element of P such that f (x) = 0 for any k-open x > x0 . Then, g(y) . f (x0 ) = y : k(y)=x0

The following proof of Theorem 7.3.2 is due to an anonymous referee. For the author’s original proof, see [Doh99f]. Proof. It suﬃces to show that y : k(y)>x0 g(y) = 0. If x0 is maximal in P , then the statement holds. We proceed by downward induction on x0 . In this way, g(y) = g(y) = f (x) = 0 , y : k(y)>x0

x>x0 x k-open

y : k(y)=x

x>x0 x k-open

90

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Combinatorial Applications and Related Topics

where the second equality comes from the induction hypothesis and the third from the hypothesis of the theorem. 2 Dualizing Theorem 7.3.2 we obtain Theorem 7.3.3 [Doh99f] Let P be a lower-finite partially ordered set, and let f and g be mappings from P into an abelian group such that f (x) = y≤x g(y) for any x ∈ P . Furthermore, let c : P → P be a closure operator and x0 a c-closed element of P such that f (x) = 0 for any c-closed x < x0 . Then, g(y) . f (x0 ) = y : c(y)=x0

From the preceding theorem we now deduce a prominent result of Rota [Rot64] on the M¨ obius function of a poset (see Deﬁnition 6.1.12), which in turn specializes to Weisner’s theorem [Wei35]. The proof is adopted from [Doh99f]. Corollary 7.3.4 [Rot64] Let P be a lower-finite partially ordered set with least element ˆ 0 and c : P → P a closure operator where c(ˆ0) > ˆ0. Then, for all x0 ∈ P , µP (y) = 0 . y : c(y)=x0

Proof. For any x ∈ P deﬁne f (x) := y≤x µP (y). There is nothing to prove if x0 is not c-closed. Otherwise, x0 = ˆ 0 and hence by (6.3), f (x0 ) = 0. Likewise, one ﬁnds that f (x) = 0 for any c-closed x < x0 . Now apply Theorem 7.3.3. 2 The preceding corollary implies one of the most important classical results of enumerative combinatorics, which is known as Weisner’s theorem [Wei35]. Corollary 7.3.5 [Wei35] Let P be a lower-finite partially ordered set with least element ˆ 0. Then, for any a > ˆ 0 and all x0 ∈ P , µP (y) = 0 . y : y∨a=x0

Proof. Deﬁne c(y) := y ∨ a for any y ∈ P and apply Corollary 7.3.4. 2

7.3.2

Application to inclusion-exclusion

We proceed by showing how the inclusion-exclusion identity (7.13) χ Av = (−1)|I|−1 χ Ai v∈V

I∈ ∗ (V ) k(I)=∅

i∈I

associated with the abstract tube of Theorem 4.2.3 may be deduced from Theorem 7.3.2. Recall that by the requirements of Theorem 4.2.3, {Av }v∈V is a ﬁnite

7.4 Matroid polynomials and the β invariant

91

family of sets, and k is

a kernel operator on V ((V ) w.r.t. Deﬁnition 7.3.1) such that x∈X Ax ⊆ v∈X / Av for any non-empty and k-open subset X of V . Proof of (7.13). For any subsets X and Y of V deﬁne   f (X) := (−1)|X| χ Ax ∩ Av ; g(Y ) := (−1)|Y | χ  Ay  x∈X

v ∈X /

y∈Y

v∈V

I:k(I)=∅

Av = Ω (the universal where, by convention, x∈∅ Ax = v∈V / set). By applying the principle of inclusion-exclusion we ﬁnd that f (X) = Y ⊇X g(Y ) for any subset X of V . Now, by the requirements, f (X) = 0 for any non-empty and k-open subset X of V . Hence, by applying Theorem 7.3.2 with P = (V ), χ Av = (−1)|I| χ Ai , from which identity (7.13) follows since χ

7.4

i∈I

Av = 1 − χ

Av . 2

Matroid polynomials and the β invariant

Our general result on sums over partially ordered sets, which we established in the previous section, is now applied to the Tutte polynomial, the characteristic polynomial and the β invariant of a matroid. In this way, some of the identities obtained for the chromatic polynomial in Section 7.2 are generalized to matroids.

7.4.1

The Tutte polynomial

In order to introduce the Tutte polynomial, we brieﬂy review some basics of matroid theory. For a detailed exposition, the reader is referred to Welsh [Wel76]. Definition 7.4.1 A matroid is a pair M = (E, r) consisting of a ﬁnite set E and a -valued function r on the power set of E such that for any A, B ⊆ E, (i) (ii) (iii)

0 ≤ r(A) ≤ |A|, A ⊆ B ⇒ r(A) ≤ r(B), r(A ∪ B) + r(A ∩ B) ≤ r(A) + r(B).

Example 7.4.2 If G is a graph and r(I) = n(G[I]) − c(G[I]) for any I ⊆ E(G), then M (G) := (E(G), r) is a matroid, which is called the cycle matroid of G. Definition 7.4.3 The Tutte polynomial T (M ; a, b) of a matroid M = (E, r) is deﬁned by T (M ; a, b) := (a − 1)r(E)−r(I)(b − 1)|I|−r(I) , I⊆E

where a and b are independent variables.

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Combinatorial Applications and Related Topics

Specializations of the Tutte polynomial T (M (G); a, b) count various objects associated with a graph G, e.g., subgraphs, spanning trees, acyclic orientations and proper λ-colorings. It is also related to the all-terminal reliability R(G). Namely, if G is a connected graph whose nodes are perfectly reliable and whose edges fail randomly and independently with equal probability q = 1 − p, then R(G) = q m(G)−n(G)+1 pn(G)+1 T (M (G); 1, q −1 ) . For further applications of the Tutte polynomial, we refer to [BO92, Wel93]. Definition 7.4.4 For any matroid M = (E, r) and any subset X of E, the contraction of X from M is deﬁned by M/X := (E \ X, rX ) where rX (I) := r(X ∪ I) − r(X) for any I ⊆ E \ X. Our result on the Tutte polynomial is stated below. An equivalent formulation using union-closed sets instead of kernel operators is left to the reader. Theorem 7.4.5 [Doh99f] Let M = (E, r) be a matroid, k a kernel operator on E and a, b ∈ such that T (M/X; a, b) = 0 for any k-open X ∈ ∗ (E). Then,

T (M ; a, b) =

(a − 1)r(E)−r(I) (b − 1)|I|−r(I) .

I : k(I)=∅

Proof. For any subset X of E deﬁne f (X) :=

(a − 1)r(E)−r(I) (b − 1)|I|−r(I) .

I⊇X

Then, by Theorem 7.3.2, it suﬃces to prove that f (X) = 0 for any non-empty and k-open subset X of E. With X denoting the complement of X in E and h(X) := (a − 1)r(E)−r(X)−rX (X) (b − 1)|X|−r(X) we obtain f (X) =

(a − 1)r(E)−r(X∪I) (b − 1)|X∪I|−r(X∪I)

I⊆X

= h(X)

(a − 1)rX (X)−rX (I) (b − 1)|I|−rX (I)

I⊆X

= h(X) T (M/X; a, b) = 0 which gives the result. 2

7.4 Matroid polynomials and the β invariant

7.4.2

93

The characteristic polynomial

From Theorem 7.4.5 we now deduce a theorem due to Heron [Her72], which generalizes Whitney’s broken circuit theorem [Whi32] from graphs to matroids. Definition 7.4.6 The characteristic polynomial C(M ; λ) of a matroid M = (E, r) is deﬁned by C(M ; λ) := (−1)r(E) T (M ; 1 − λ, 0) = (7.14) (−1)|I| λr(E)−r(I) . I⊆E

A circuit of a matroid M = (E, r) is set C ∈ (E) such that r(C \ {c}) = |C| − 1 = r(C) for any c ∈ C. A loop is a circuit of cardinality 1. If E is linearly ordered and C a circuit of M , then C \ {max C} is called a broken circuit of M . ∗

Remark. By the classical inclusion-exclusion principle, the chromatic polynomial of a graph G is related to the characteristic polynomial of M (G) by PG (λ) = λc(G) C(M (G); λ) . We are now ready to state Heron’s broken circuit theorem [Her72]. The proof is adopted from [Doh00d]. Theorem 7.4.7 [Her72] Let M = (E, r) be a matroid, where E is endowed with a linear ordering relation. Then,

r(E)

C(M ; λ) =

(−1)k bk (M ) λr(E)−k

k=0

where bk (M ) is the number of k-subsets of E which do not include a broken circuit of M as a subset. Proof. By Lemma 1.4 of Heron [Her72] we have (7.15)

C(M ; λ) = 0 if M contains a loop.

For any X ⊆ E deﬁne k(X) :=

{C ⊆ X | C is a broken circuit of M }.

Then, for any non-empty and k-open X ⊆ E there is some e > max X such that X ∪ {e} includes a circuit of M . Therefore, r(X ∪ {e}) = r(X), or equivalently, rX ({e}) = 0. From this we conclude that e is a loop of M/X and hence by (7.15), C(M/X; λ) = 0. By applying Theorem 7.4.5, using identity (7.14) and the fact that r(I) = |I| for any I including no broken circuit of M , we obtain C(M ; λ) = (−1)|I| λr(E)−|I| , I⊆E I⊇X(∀X∈

)

where is the set of broken circuits of M . 2

94

7

7.4.3

Combinatorial Applications and Related Topics

The β invariant

Results similar to Theorem 7.4.5 and Theorem 7.4.7 can also be established for Crapo’s β invariant [Cra67], which among other things indicates whether M is connected and whether M is the cycle matroid of a series-parallel network. Definition 7.4.8 The β invariant of a matroid M = (E, r) is deﬁned by (7.16) β(M ) := (−1)r(E) (−1)|I| r(I) . I⊆E

A matroid M = (E, r) is called disconnected if there is a pair of distinct elements of E that are not jointly contained by a circuit of M . Remark. By Lemma II of Crapo [Cra67], β(M ) = 0 if M is disconnected or a loop. This will be used in the proof of the following theorem. Theorem 7.4.9 [Doh99f] Let M = (E, r) be a matroid, and let k be a kernel operator on E such that E is not k-open and M/X is disconnected or a loop for any non-empty and k-open subset X of E. Then, β(M ) = (−1)r(E) (−1)|I| r(I) . I : k(I)=∅

Proof. By the assumptions of the theorem and the preceding remark, β(M/X) = 0 whenever X ⊆ E is non-empty and k-open. For any subset X of E deﬁne f (X) := (−1)|I| r(I) . I⊇X

Then, by Theorem 7.3.2 it suﬃces to show that f (X) = 0 for any non-empty and k-open subset X of E. By the assumptions, X is non-empty and therefore, . . (−1)|X∪I| r(X ∪ I) = (−1)|I| (rX (I) + r(X)) = β(M/X) = 0, f (X) = I⊆X

I⊆X

. where a = b means that a = b or a = −b. 2 As a corollary we obtain the following well-known result: Corollary 7.4.10 Let M = (E, r) be a matroid, where E is endowed with a linear ordering relation. Then, β(M ) = (−1)

r(E)

|E|

(−1)k k bk (M )

k=1

where again bk (M ) is the number of k-subsets of E including no broken circuit. Proof. Corollary 7.4.10 follows from Theorem 7.4.9 in the same way as Theorem 7.4.7 follows from Theorem 7.4.5. 2

7.5 Euler characteristics and M¨obius functions

7.5

95

Euler characteristics and M¨ obius functions

In this section, our general result on sums over partially ordered sets, which we established in Theorem 7.3.2, is applied to the Euler characteristic of an abstract simplicial complex and to the M¨obius function of a ﬁnite lattice.

7.5.1

Euler characteristics

Our result on the Euler characteristic requires the following deﬁnition. Definition 7.5.1 For any abstract simplicial complex and any simplex X ∈ the link /X is the abstract simplicial complex deﬁned by /X := {I ∈ | I ∩ X = ∅, I ∪ X ∈ } . Theorem 7.5.2 Let k be a kernel operator on the vertex-set of an abstract simplicial complex such that γ( /X) = 1 for any k-open X ∈ . Then, γ( ) = γ ({I ∈ | k(I) = ∅}) . Proof. Let

+

:=

f (X) :=

∪ {∅}, and for any subset X of the vertex-set of

g(Y ), where

g(Y ) :=

Y ⊇X

(−1)|Y |−1 0

deﬁne

if Y ∈ + , otherwise.

Then, f (X) = 0 for any non-empty X ⊆ Vert( ) which is not a simplex of , and f (X) = (−1)|X|−1 +

(−1)|Y |−1 = (−1)|X|−1 + (−1)|X| γ( /X) = 0

Y∈ Y ⊃X

for any k-open simplex X ∈ . Hence, by applying Theorem 7.3.2, I∈

+

(−1)|I|−1 =

(−1)|I|−1 .

I∈ + k(I)=∅

Now, by subtracting the term for I = ∅ from both sides the result follows. 2 In view of Theorem 7.5.2 we pose the following conjecture. For the notion of homotopy equivalence, the reader is referred to the textbook of Harzheim [Har78]. Conjecture 7.5.3 Let k be a kernel operator on the vertex-set of an abstract simplicial complex such that /X is contractible for any k-open simplex X ∈ . Then, the complexes and {I ∈ | k(I) = ∅} are homotopy equivalent.

96

7.5.2

7

Combinatorial Applications and Related Topics

M¨ obius functions

Our result on M¨ obius functions, stated as Theorem 7.5.8 below, is an extension of Rota’s crosscut theorem [Rot64] as well as a generalization of a result due to Blass and Sagan [BS97]. We need the following notions due to Rota [Rot64]. Definition 7.5.4 Let L = [ˆ 0, ˆ 1] be a ﬁnite lattice. L is called non-trivial if ˆ ˆ L \ {0, 1} = ∅. An element a ∈ L is an atom of L if a > ˆ0 and no a ∈ L satisﬁes a > a > ˆ 0. The set of atoms of L is denoted by A(L). A crosscut of L is an antichain C ⊆ L\{ˆ 0, ˆ 1} having a non-empty intersection with any maximal chain ˆ ˆ from 0 to 1 in L. The crosscut complex of L associated with C is deﬁned by Γ(L, C) := I ∈ ∗ (C) I > ˆ0 or I < ˆ1 and is easily seen to be an abstract simplicial complex. For any abstract simplicial complex , we refer to γ˜ ( ) := γ( ) − 1 as the reduced Euler characteristic of . Example 7.5.5 A(L) is a crosscut for any non-trivial ﬁnite lattice L. The following proposition, which is stated without proof, is known as Rota’s crosscut theorem [Rot64]. A proof of Rota’s crosscut theorem can also be found in the textbook of Aigner [Aig79]. In the following, we write µ(L) instead of µL (ˆ 1), where µL denotes the M¨obius function of L (see Deﬁnition 6.1.12). Proposition 7.5.6 [Rot64] Let L = [ˆ 0, ˆ 1] be a non-trivial finite lattice, and let C be a crosscut of L. Then, µ(L) equals the reduced Euler characteristic of the crosscut complex of L associated with C, or equivalently, (7.17) µ(L) = (−1)|I| . ∗ Ï(C) ÎI∈ I=ˆ 0, I=ˆ 1 Example 7.5.7 Let L be the lattice shown in Figure 7.2. Evidently, C = {a, b, c, d, e} is a crosscut of L. The associated crosscut complex Γ(L, C) equals {{a}, {b}, {c}, {d}, {e}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {b, e}, {c, d}, {c, e}, {d, e}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}, {a, b, c, d}, {b, c, d, e}} and contains 23 simplices each of which contributes to the reduced Euler characteristic γ˜ (Γ(L, C)) = µ(L). Alternatively, we can compute µ(L) via (7.17), which gives a sum involving only eight terms. Anyway, µ(L) = 0. Our result on M¨ obius functions shows that µ(L) may be considered as the reduced Euler characteristic of a subcomplex of Γ(L, C).

7.5 Euler characteristics and M¨obius functions

97

1

a

b

c

d

e

0

Figure 7.2: A lattice with crosscut {a, b, c, d, e}.

ˆ 1] ˆ be a non-trivial finite lattice, and let C be a Theorem 7.5.8 Let L = [0, crosscut of L, which is endowed with a partial order which is denoted by to distinguish it from the partial order ≤ in L. Furthermore, let k be a kernel operator on C such that for any non-empty and k-open subset X of C and any x ∈ X there is some c ∈ C satisfying c x and X < c < X. Then, (7.18)

µ(L) = γ˜ ({I ∈ Γ(L, C) | k(I) = ∅})

or equivalently, (7.19)

µ(L) =

Î Ï

(−1)|I| .

I∈ ∗ (C) I=ˆ 0, I=ˆ 1 k(I)=∅

Proof. By Theorem 7.5.2 and Proposition 7.5.6, (7.18) is proved if (7.20)

γ (Γ(L, C)/X) = 1

for any k-open simplex X ∈ Γ(L, C). We ﬁrst observe that (7.21) Γ(L, C)/X = I ∈ ∗ (C \ X) (I ∪ X) > ˆ0 or (I ∪ X) < ˆ1 . Now, let cX be -minimal | X < c < X}. Then, cX ∈ / X since in {c ∈ C otherwise c cX and X < c < X for some c ∈ C, contradicting the minimality of cX . Now, by (7.21) and since X < cX < X and cX ∈ / X, it

98

7

Combinatorial Applications and Related Topics

follows that Y → Y {cX } is a sign-reversing involution on Γ(L, C)/X ∪ {∅}, whence (7.20) and thus (7.18) is shown. To establish (7.19) we ﬁrst note that (7.22) γ˜ ({I ∈ Γ(L, C)|k(I) = ∅}) = γ˜({I ∈ ∗ (C)|k(I) = ∅}) + (−1)|I| . ∗ Ï(C) ÎI∈ I=ˆ 0, I=ˆ 1 k(I)=∅

The assumptions of the theorem imply that C is not k-open, whence = ∗ (C) satisﬁes the requirements of Theorem 7.5.2. Thus, by applying Theorem 7.5.2, γ˜({I ∈ ∗ (C) | k(I) = ∅}) = γ˜(∗ (C)) = 0 .

(7.23)

In view of (7.22) and (7.23) the equivalence of (7.18) and (7.19) is obvious. 2 Corollary 7.5.9 Let L = [ˆ 0, ˆ 1] be a non-trivial finite lattice and C be a crosscut of L which is given a partial order denoted by to distinguish it from the partial order ≤ in L. Let consist of all non-empty subsets X of C such that for any x ∈ X there is some c ∈ C satisfying c x and X < c < X. Then, (7.24)

µ(L) = γ˜ ({I ∈ Γ(L, C) | I ⊇ X for any X ∈ })

or equivalently, (7.25)

µ(L) =

Î Ï

(−1)|I| .

I∈ ∗ (C) I=ˆ 0, I=ˆ 1 I⊇X(∀X∈ )

Proof. Since turns out as union-closed, the corollary follows from Theorem 7.5.8 and the correspondence between kernel operators and union-closed sets. 2

Example 7.5.10 Let L and C be as in Example 7.5.7. By putting d b, d e, b a and b c a partial ordering relation on C is deﬁned, whose Hasse diagram is shown in Figure 7.3. In connection with this partial ordering relation, = {{a, c}, {b, e}, {c, e}} satisﬁes the requirements of Corollary 7.5.9, and thus µ(L) can be expressed via (7.24) as the reduced Euler characteristic of the subcomplex {{a}, {b}, {c}, {d}, {e}, {a, b}, {a, d}, {b, c}, {b, d}, {c, d}, {d, e}, {a, b, d}, {b, c, d}} of Γ(L, C), which contains only 13 of the total 23 simplices of Γ(L, C). Moreover, for the present choice of it turns out that the sum on the right-hand side of (7.25) contains only two of the eight terms that appear in the sum of (7.17).

Remarks. In view of Theorem 7.5.8 the corollary can be stated more generally by requiring to be a union-closed set of non-empty subsets of the above type.

7.5 Euler characteristics and M¨obius functions

a

99

c

b

e

d

Figure 7.3: Hasse diagram of {a, b, c, d, e} with respect to .

By putting C = A(L) Corollary 7.5.9 specializes to a result of Blass and Sagan [BS97], which they used in computing and combinatorially explaining the M¨obius function of various lattices and in generalizing Stanley’s well-known theorem [Sta72] that the characteristic polynomial of a semimodular supersolvable lattice factors over the integers. As pointed out by Blass and Sagan [BS97], their result generalizes a particular case of Rota’s broken circuit theorem [Rot64] as well as a prior generalization of that particular case due to Sagan [Sag95]. For a restatement of Rota’s result [Rot64], the following deﬁnitions are necessary: ˆ 1] ˆ be a ﬁnite lattice. A rank function of L is a Definition 7.5.11 Let L = [0, function r : L → ∪ {0} such that r(ˆ 0) = 0 and r(a) = r(b) + 1 whenever a is an immediate successor of b. It is well known and straightforward to prove that the rank function is unique if it exists. A geometric lattice is a ﬁnite lattice L whose rank function exists and satisﬁes r(a ∧ b) + r(a ∨ b) ≤ r(a) + r(b) for any a, b ∈ L. A subset I of A(L) is independent if r( I) = |I|, and dependent otherwise. Each minimal dependent subset of A(L) is called a circuit of L. Given a linear ordering relation on A(L), each set C \ {max C}, where C is a circuit of L, is referred to as a broken circuit of L. Note that this depends on the ordering of the atoms. Corollary 7.5.12 [Rot64] Let L = [ˆ 0, ˆ 1] be a non-trivial geometric lattice with rank function r. Furthermore, let the set of atoms A(L) be endowed with a linear ordering relation, and let consist of all broken circuits of L. Then, ˆ µ(L) = (−1)r(1) × I ⊆ A(L) I = ˆ 1 and I ⊇ B for any B ∈ . Proof. [BS97] (Sketch) Let C = A(L) and as in Corollary 7.5.9. Then, ⊇ and any X ∈ includes some B ∈ . Thus, for any I ⊆ A(L), I ⊇ X for any X ∈ if and only if I ⊇ B for any B ∈ . In this case, I is independent and hence r( I) = |I|. Thus, Corollary 7.5.12 follows from Corollary 7.5.9. 2

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Author Index Alexandroﬀ, 22 Ball, 44, 49, 56, 75, 77, 78 Barlow, 68 Bernoulli, 2 Birkhoﬀ, 83, 84 Bj¨orner, 21 Blass, 99 Bonferroni, 2 Boole, 2 Borsuk, 15 Buksz´ar, 37 Buzacott, 56, 57 Colbourn, 47, 55 Crapo, 94 Da Silva, 2 Dawson, 64, 65, 79, 81 Edelman, 20, 21, 31, 32 Edelsbrunner, 14 Galambos, 2, 37, 38, 41, 42 Garsia, 33 Giglio, 81 Gordon, 32 Heidtmann, 68 Heron, 93 Howorka, 23 Hunter, 37, 39 Hwang, 73 Jamison, 20, 22, 31 Jordan, 2 Kossow, 71, 74 Kounias, 42 Kwerel, 42, 64, 65, 79, 81 Lawrence, 32 Lozinskii, 3 Manthei, 52, 53

McIlwain, 74 McKee, 37 Milne, 33 Montmort, 1 Naiman, 4, 9–18, 23, 81 Narushima, 4, 22, 25, 31, 82, 83 Poincar´e, 1 Pr´ekopa, 37 Prabhakar, 52 Preuss, 71, 74 Provan, 44, 49, 54, 56, 75–78 Pr´ekopa, 64 Ramos, 14 Read, 55 Reiner, 21, 32 Rota, 90, 96, 99 Sagan, 99 Sankoﬀ, 64, 65, 79, 81 Satyanarayana, 52 Seneta, 41 Shanthikumar, 73 Shier, 4, 44, 47, 49–51, 54–56, 71, 74–78 Simonelli, 2, 37, 38, 41 Stanley, 86, 99 Sylvester, 1 Tak´ acs, 2 Tarjan, 55 Tittmann, 70 Tomescu, 37 Weisner, 90 Whitney, 4, 83, 84, 86 Wilf, 83, 84 Worsley, 37, 39 Wynn, 4, 9–18, 23, 81 Xu, 42 Zeilberger, 4, 28 Ziegler, 21

Subject Index abstract lace expansion, 28 abstract simplicial complex, 7 abstract tube, 9–18 all-terminal reliability, 54

importance sampling scheme, 13 inclusion-exclusion principle, 1

basis, 20 beta invariant, 94 betti numbers, 11 binomial moments, 64 broken circuit, 84, 93, 99 broken circuit complex, 84

lace map, 28

characteristic polynomial, 93 chordal graph sieve, 38 chromatic polynomial, 84 clique complex, 23 clique number, 5 closed, 20, 89 closure operator, 20, 89 coherent binary system, 47 contractible, 8 convex geometry, 20 convex polyhedron, 14 crosscut complex, 96

nerve, 10 network reliability, 53–66

Delauney complex, 16, 18 depth of an abstract tube, 13 domination, 31, 52 dual antimatroid, 20 dynamic programming, 45, 50 Euler characteristic, 7, 96 Euler-Poincar´e formula, 11 extended set of mincuts, 51 extended set of minpaths, 51 free set, 20

kernel operator, 25, 89

M¨ obius function, 52 matroid, 91 method of indicators, 1 mincut, 48 minpath, 48

open, 25, 89 order complex, 22 poset terminology, 6 pseudopolynomial algorithm, 51 quasi-cut, 57 recursive scheme, 44 reliability covering problems, 75 semilattice sieve, 22 source-to-T -terminal reliability, 54 source-to-terminal reliability, 54 subtube of an abstract tube, 9 system reliability, 48 Tutte polynomial, 91 two-terminal reliability, 54 unions of balls, 15 unions of spherical caps, 17 Voronoi subdivision, 16, 18

generalized chromatic polynomial, 86 graph terminology, 5 Helly number, 22 homology group, 11

weak abstract tube, 10