(0, 1) ideal matrices

Mathematical Programming 80 (1998) 265-281

(0, + 1 ) ideal matrices 1 Paolo Nob±l± a, Antonio Sassano b,, a lstituto di Anal±s±de± Sistemi ed lnformatica del CNR, Viale Manzoni 30, 00185 Rome, Italy b Dipartimento di lnformatica e Sistemistica, Universitgt di Roma "La Sapienza ", via Buonarroti, 12, 00185 Rome, Italy Received 1 February 1995; revised manuscript received 12 February 1996

Abstract A (0, 1) matrix A is said to be ideal if all the vertices of the polytope Q ( A ) = {x [ Ax ) 1, 0 ~< x ~< 1} are integral. The issue of finding a satisfactory characterization of those matrices which are minimally non-ideal is a well known open problem. An outstanding result toward the solution of this problem, due to Alfred Lehman, is the description of crucial properties of minimally non-ideal matrices. In this paper we consider the extension of the notion of ideality to (0, ± 1 ) matrices. By means of a standard transformation, we associate with any (0, ± 1 ) matrix A a suitable (0, 1 ) matrix D ( A ) . Then we introduce the concept of disjoint completion A + of a (0, ± 1 ) matrix A and we show that A is ideal if and only if D ( A +) is ideal. Moreover, we introduce a suitable concept of a minimally non-ideal (0, ± 1) matrix and we prove a Lehman-type characterization of minimally non-ideal (0, ±1 ) matrices. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Keywords: Generalized set covering problem; Perfect and ideal matrices; Lehman's theorem; (0, 4-1 ) matrices

1. Introduction Let A be a (0, + 1 ) matrix w h o s e sets o f c o l u m n s and rows are, respectively, N = { 1 . . . . . n} and M = { 1 . . . . . m}. W e define the generalized set covering p o l y t o p e associated w i t h the matrix A as the solution set Q ( A ) o f the f o l l o w i n g system:

~ - ~ x j - ~-~ xj >~ l - Iei I, jCPI

i 6 M,

jCRI

0 ~ < x j ~< 1,

( 1) j6N,

* Corresponding author. 1This work was partially supported by MURST, Rome, Italy. 0025-5610/98/$19.00 @) 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. PH S0025-56 1 0 ( 9 7 ) 0 0 0 0 8 - 7

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where, for each row i of the matrix A, Pi is the set of columns which have a +1 entry in row i and Ri is the set of columns which have a - 1 entry in row i. Evidently, Pin R i = ~. With a slight abuse of notation, we will sometime use the term "row i" referring to the set Pi U gi, thus allowing statements like "row i contains column j " or "row i intersects row k". The matrix A is said to be the constraint matrix of the system (1). Let B be any submatrix of A. The restriction of a row i of A to the matrix B is the intersection of i with the column set of B. A column of A is said to be monotone if it does not contain both a 1 and a - 1. The matrix A / obtained from A by multiplying by - 1 all the entries of a column j is said to be a switching of A. We also say that A / is obtained by switching column j. Observe that switching column j corresponds to replacing variable xj by variable yj = 1 - xj in (1). A zero vector, denoted by 0, is a vector with all components equal to zero. A one vector, denoted by 1, is a vector with all components equal to 1. m m In what follows we will denote by P the set Ui=l Pi, by R the set Ui=l Ri and by u ( A ) the vector whose ith component is IRil. This notation will allow us to write system (1) in the more compact form

Ax ~ 1- u(A), 0~<x~
(ii) the contraction of j in A is the submatrix A / { j } obtained by removing all rows with a - 1 in column j, removing column j and, finally, removing any dominated row.

Observe that deletion and contraction as defined above, generalize the corresponding operations for (0, 1) matrices and so we use the standard notation for them. Deletion and contraction operations are commutative. Hence, for each pair C, D of subsets of N such that C n D = ~, we can contract the columns in C and delete the

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columns in D in any order, obtaining the submatrix A ~ = A \ D / C of A. The matrix A ~ is said to be a minor of A. Moreover, as in the (0, 1) case, the system ( a \ { j } ) x ~> 1 - ~ , ( a \ { j } ) is obtained from ax >1 1 - p ( a ) by fixing at 1 the variable xj. Similarly, the system ( a / { j } ) x >1 1 u ( A / { j } ) is obtained from Ax ~> 1 - ~,(A) by fixing at 0 the variable xj. Consequently, again as in the (0, l) case, it can be shown that every minor of an ideal matrix is ideal. Following Guenin [3] we define also an elimination operation on a (0, 4-1) matrix A, as follows: Definition 2. Let A be a (0, 4-1) matrix and j a column of A. Then the elimination of j in A is the submatrix A - {j} obtained by removing all rows with a non-zero entry in column j and removing column j. Unfortunately, the property of idealness is not closed under elimination. In other words, it may happen that an ideal matrix is transformed into a non-ideal one by eliminating one column. Let i, k be two rows in M such that Pin Rk = {l} and Pk N Ri = ~. The following constraint is said to be an implication of the system ( 1 ) :

xi jC(PiUPk-{l})

-

Z

x;/> a -

IRe U Rk -

{Z}l.

(2)

jC(RiURk-{l})

Observe that the constraint (2) has the same structure as the constraints in (1). Rows i and k are said to be the antecedents of (2). If (Pi U Ri) n (Pk U R~) = {l} we say that (2) is a disjoint implication. Observe that in this case the constraint (2) is a linear consequence of system (1). The system obtained starting from ( 1 ), recursively adding all implications and finally removing all dominated rows is said to be the logical completion of (1). The constraint matrix of such a system is also called the logical completion of A and denoted as A •. Since any constraint of the form (2) is satisfied by every (0, 1) vector x which satisfies its antecedents i and k, we have that Q(A ~) n {0, 1} n = Q(A) n {0, 1} n. Analogously, the system obtained by recursively adding all disjoint implications and removing all dominated constraints is called the disjoint completion of (1). Its constraint matrix is called the disjoint completion of A and denoted as A+; evidently, Q(A) = Q(A+). A matrix A which coincides with its disjoint completion will be called d-

complete. In the rest of the paper we will refer to a row of A associated with a dominated constraint (with an implication) as a dominated row (an implication). The main purpose of this paper is to establish a connection between the properties of (0, 4-1) ideal matrices and those of (0, 1) ideal matrices. Such a connection will allow us to define a sound concept of minimally non-ideal matrix for the (0, 4-1) case and to give a characterization of the minimally non-ideal (0, 4-1) matrices. A crucial step to achieve this goal is the description of a standard transformation which maps the polytope Q(A) associated with a (0,4-1) matrix A to a face of a

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suitable set covering polytope Q(B), associated with a (0, 1) matrix B. To describe such a standard transformation, we first rewrite the system (1) as follows:

Zxj+ jEPi

(1

xj) >l, icM, (3)

jERi

O~x.i<~ l,

jEN.

Now, if we let y / = xj for each j E P and zj = 1 - xj for each j E R, we can rewrite system (3) as follows:

Zyj+ZZy>~I, jEPi

iEM,

jERi

yj + z j = 1,

jEPnR,

39>>.0,

j E P,

zj>/O,

j ER.

(4)

I_~t D(A) be the incidence matrix of the constraints in (4). If d = IP n RI is the number o f non-monotone columns o f A, then D ( A ) is a (0, 1 ) matrix with m + d rows and n + d columns. The first m rows of D ( A ) correspond to the constraints o f ( 1 ), the remaining d rows correspond to the coupling constraints yj + zj = 1 in (4). The set o f columns of D ( A ) is partitioned into two sets: Each column in the first set corresponds to a variable yj (j E P) and is derived from the j t h column of A by turning to zero the - 1 entries. Each column in the second set corresponds to a variable zj (j E R) and is derived from the j t h column o f A by first switching it and then by turning to zero the - 1 entries. To make clear the connection of a column o f D(A) with the column of A from which it derives, we will call positive (negative) the columns in the first (second) set. For a set o f columns J C N we shall denote by p ( J ) and r(J) the corresponding subsets o f positive and negative columns o f D ( A ) , with p ( J ) = (~ if J N P = ~, r ( J ) = if J n R = (~, p ( j ) = p ( { j } ) and r(j) = r ( { j } ) . As one can promptly see, the system (4) defines a face F(A) of the set covering polytope Q ( D ( A ) ) . Moreover, there is a one-to-one correspondence between solutions o f (1) and solutions o f (4). In particular, if P = N then such a correspondence is defined by y = x, z = 1 - x, implying that Q(A) is the projection onto R n of F(A). We call D(A) the (0, 1)-extension of A. Moreover, if x is a vector in Q(A), we call extension of x the vector ( y , z ) E Q ( D ( A ) ) , where Yi = x i , for i E P and zi = 1 - x i , for i E R.

2. A characterization of ideal (0, 4-1) matrices In [4] John Hooker defines a set covering submatrix of a ( 0 , ± 1 ) matrix A as a maximal row submatrix such that every column is monotone. Using such a concept, he was able to give the following characterization o f the ideal (0, 4-1) matrices that coincide with their logical completion.

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Theorem 3 (Hooker). If A ~ A ° then Q(A) is integral if and only if, for every set

covering submatrix B of A, Q(B) is integral. In the following theorem, we give a more general characterization of ideal (0, ±1) matrices. Theorem 4. Let A be a (0, ~1) matrix. Then A is ideal if and only if D(A +) is ideal. Proof. ("if") Assume that A is non-ideal, that is, there exists some non-integral vertex 2 of Q(A) =_Q(A+). Let (9, z) be the extension of ~. Obviously, (9, z) is an extreme point of the face F (A + ) of Q ( D ( A +) ) and, hence, it is an extreme point of Q ( D ( A + ) ). This proves that D(A +) is non-ideal. ("only if") Assume now that D(A +) is non-ideal, that is, there exists some nonintegral vertex (9, z) of Q(D(A ÷)). Consider first the case A ÷ = A% If (Y, z) C F(A) then 9 is a (non-integral) vertex of Q(A), proving that A is nonideal. Assume, conversely, that (9, z) ~ F(A). Hence there exists a column h E N such that 9h + ~h > 1.

(5)

Since (9, Z) is an extreme point of Q(D(A +)), Yh and ~h cannot be decreased without violating some constraint defining Q(D(A +) ). In other words, there exist two rows u and v of D(A +) which satisfy: Zgj+E~j=I, .jCP,,

~-~ 9.i + ~ .jEPv

(6)

.jER,,

Zj= 1,

(7)

jGRv

with h E P u n Rv. Let P' = (Pu UPv) - (R. U R~) and R' = (R. U R~) - (Pu U P~). By adding (6) and (7) we obtain

EgJ+ jGP'

J+ E jER'

jGPuAR,~

(8) jGPvNR.

Since Yj + zj ~> 1 for any j E N, we have Pu N R, = (h) and P, N R, = •. Hence, by definition of logical completion, there exists some row in D(A °) = D(A +) whose sets of positive and negative columns are contained in pi and R I, respectively. If P~ U R t = O, D(A ~') contains the empty row and hence Q(D(A°)) is empty, contradicting the hypothesis that it contains the vertex (:9, ~). It follows that P~ U R~ ¢ 0 and that the inequality

Zyj+~zj>~l, jCP'

(9)

jGR t

is satisfied by all the points in Q(D(A+)) = Q(D(A°)). On the other hand, by subtracting (5) from (8), it follows that (9) is not satisfied by (9, z), a contradiction.

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Assume now that A ° ~ A +.

Claim 1. If A ~ 4= A + then Q ( A ~) C Q ( A + ) . We prove the claim by exhibiting a fractional point 2 in Q ( A +) - Q(AO). Since A ~ v~ A + we have that there exist rows of A ° that are not rows o f A +. Let w be one o f these rows such that k = IPw U Rwl has the minimum value. By switching columns in Rw, we can assume, without loss of generality, that Rw = O. Let A ° be the submatrix o f A + obtained by elimination of the columns in C o = N - P w . Let 2 o be the vector of Q ( A °) whose components are all equal to 1 / ( k + 1). Observe that 2 ° violates the constraint

xj >~ 1,

(10)

.jc=Pw

associated with row w of A ~. However, we claim that 2 ° satisfies the constraint defined by any row u o f A °. In fact, we have that ]R. ] ~> 2, otherwise either u or the implication of u and w would dominate row w in A °. The inequality associated with row u can be written as follows:

~-~x.j + ~--~(1 - - x j ) >~ 1. jCP,,

(11)

jGR,,

Since JR,] ~> 2 and each component of 2~° has value 1 / ( k + 1), we have that (11) is satisfied by 2 o . Hence, we have that the vector 2 ° is in Q ( A °) ~ Q ( A + - C °) but violates inequality (10). Moreover, .~0 satisfies the following property: (,)

Every component has a value between 1/ (k + 1 ) and k~ (k + 1 ).

Let C ~ be the smallest subset o f C O such that there exists a vector 2~ ~ Q ( A + - C ~) which violates inequality (10) and satisfies property ( , ) . The theorem will be proved by showing that C ~ = ~. Assume, conversely, that ]C'] /> 1, let h be an index in C ' and C " = C ' - {h}. To complete the p r o o f we now show that there exists a fractional point Yc" E Q ( A + C ' ) which violates inequality (10) and satisfies property ( * ) , thus contradicting the assumption of minimality of C t. To this purpose, let 2 " be defined as follows:

2~t=~,

j6

(N-C'),

Evidently, for any choice of a the vector :~" violates inequality (10). We will show that the value ae can be determined in such a way that 1 / ( k + 1) ~< a ~< k / ( k + 1 ) and every constraint associated with a row o f A " = A + - C " is satisfied by .~". If this is not the case, then there exist two rows r and s o f A " (and hence of A +) whose associated constraints can be written as follows:

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_xj + Z ( 1 - xj) >1 1, .jEPr

.jCRr

ZXj

+ Z(1

jCP,

271

(12)

--Xj) >~ 1,

(13)

jCRs

such that h E Pr MRs and, for any value ~ E [ 1/ (k + 1 ), k~ (k + 1) ], at least one of the two constraints is violated by 2". Observe that ]Ps tORs ] ~> 2 and ]Pr tORrl ~> 2, otherwise the disjoint implication of r and s would dominate both r and s in A +. Consequently, by property (*), letting o~ = 1 / ( k + 1) the vector 2~' satisfies inequality (13) and, hence, violates inequality (12). On the other hand, letting ~ = k / ( k ÷ l) the vector 2 ~' satisfies inequality (12). It follows that there exists a value & E [ 1 / ( k + 1 ) , k / ( k ÷ 1)] such that, letting ~ = &, the vector 2" satisfies: ~-~ 2~! + ~ ( 1 jEPr

27 + jCP~

- 2y) = 1,

(14)

27) < 1.

(15)

jERr

jGRs

Let P ' = Pr UPs - {h} and R' = Rr tJ Rs - {h}. Summing up inequalities (14) and (15) and observing that xj -" = 2~, for j 4; h, we get the following relation: Z2;+S(1-2~)+ .jCP'

jCR'

~ jGPrNP,

2;+

~

(1-2;)<1.

(16)

jERrV)R,

Observe that P~ N R ~ = 0, otherwise the inequality (16) would be contradicted. It follows that rows r and s are the antecedents of an implication. Suppose now that there exists some row t in A + such that Pt C_ P~ and Rt C R I. Observe that t is also a row in A". Hence the inequality

xj + ~ ( 1 jGP t

- xj) ~> 1,

(17)

jCzRt

is satisfied by all the points in Q ( A " ) , contradicting inequality (16). Consequently, rows r and s are the antecedents of a non-disjoint implication which is not dominated by any row in A +. Then there exists some row t of A ~ which is not in a + such that Pt C_ p' and R t C R t. Since ]Pt to Rt] ) k, we have IP ' U R' I ) k. Moreover, we have [(Pr f3 Ps) U (Rr n R~)I/> 1. Hence, by property ( , ) we get again a contradiction to inequality (16). We can conclude that there exists a value ce c [ 1/ (k + 1), k / ( k + 1) ] such that the point 2" belongs to Q(A") and violates inequality (10).

Claim 2. If A ~ # A + then A is non-ideal. Observe that, by Claim 1, Q(A ~) c Q(A+). Moreover, Q(A °) N{0, 1} n = Q ( A +) fq {0, 1} n, and so Q(A +) - Q(A) has a fractional vertex. Consequently, A is non-ideal. Claim 2 concludes the proof of the theorem.

[]

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Recently [3], Guenin was able to prove a related result. Given a (0, ± 1 ) matrix A he defined the matrix A* to be the matrix obtained from A by adding any (0, 4-1) row such that the associated constraint of the form ( 1 ) is valid for Q (A).

T h e o r e m 5 (Guenin). Let A be a (0, -4-1) matrix. Then A is ideal if and only if D ( A * ) is ideal. In general A + v~ A*, as the following example shows.

A=

1

0 1

0 0

1 1

0

1

0

"

(18)

Observe that A ~ A + ¢ A*, since the row (0 1 1 1 1) belongs to A* but not to A +. Evidently, Q ( D ( A ) ) D_ Q ( D ( A + ) ) D Q ( D ( A * ) ) 2 Q ( D ( A ~ ) ) . Moreover, if A is ideal then, by Claim 2 of Theorem (4), we have that A + = A* = A ° and so D ( A +) = D ( A * ) = D ( A * ) . Then, by Theorem 4, D ( A * ) is ideal. Conversely, if A is non-ideal then F ( A ) =-- F ( A +) ~ F ( A * ) has a fractional vertex, and so D ( A * ) is non-ideal. It follows that Guenin's theorem is a corollary of Theorem 4. On the contrary, Hooker's theorem (Theorem 3) is not an immediate consequence of Theorem 4. In fact, the discussion on the relationship between our result and Hooker's theorem requires some more definitions and will be one of the subjects of the next section.

3. M i n i m a l l y non-ideal (0, -4-1) matrices A (0, 1) matrix is said to be minimally non-ideal if it is non-ideal but every proper minor is ideal. Alfred Lehman [5] showed that minimally non-ideal (0, 1) matrices have certain regularity properties except for one infinite family. In order to describe this result, we need to introduce some definitions. We denote by Js a square ( s + l × s + l ) (0, 1) matrix whose rows are the characteristic vectors over the set {1 . . . . . s + 1} (s ~> 2) of the following sets: {2, 3 . . . . . s + 1}, {1, i} for i = 2 . . . . . s + 1. Moreover, given a (0, 1) (m x n) matrix A, let ai denote the ith row of a and let p ( a ) = min{aT1 [ i = 1 . . . . . m}. Let / 3 ( a ) denote the minimum of xWl over all the (0, 1) vectors in Q ( A ) (covering number of A). Finally, let E be the matrix of all ones and I the identity matrix of appropriate dimension. T h e o r e m 6 (Lehman). If A is a minimally non-ideal (0, l) matrix with n columns, then either (a) A is J,,-1; or (b) there are exactly n linearly independent rows of A, rl . . . . . rn, with p ( A ) ones and n (0, 1) linearly independent vectors o f Q( A ) , b~ . . . . . bn, with fl( A ) ones.

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Moreover, denoting by R and B the square matrices whose rows are the vectors ri and bi, respectively, then the following hold: ( b l ) RB T = E + ( p ( A ) f l ( A ) - n ) I ; (b2) each column of R has p ( A ) ones and each column of B has f l ( A ) ones. An easy corollary of Lehman's theorem is the existence of a unique fractional vertex of Q (A) when A is minimally non-ideal. Corollary 7. If A is minimally non-ideal, then Q( A ) has a unique fractional vertex Yc. In particular, if A = Jn-1 then £1 = ( n - 2 ) / ( n - 1), -~i = l / ( n - 1), for i= 2 . . . . . n. If, conversely, A ~ Jn-1 then xi - = 1 / p ( A ) , f o r i = 1 . . . . . n. It is natural to extend the notion of a minimally non-ideal matrix to the (0, =El) case. In particular, we would like to define an appropriate generalization of this notion which preserves the nice regularity properties expressed by Lehman's result. However, if we define a minimally non-ideal matrix A as a matrix which is non-ideal but all of whose proper minors are ideal, we loose the fundamental property that Q ( A ) has a unique fractional vertex. In fact, the following matrix M has the property that any deletion or contraction produces an ideal submatrix, but Q ( M ) has two fractional vertices, namely, £1 = (3, ~,g)l and 2~2 = (1, ~, 32_).

M=

111:] 1

1

1

0

1

-1

-1 -1

-1 0

o1

(19)

In [3] Bertrand Guenin shows that deletion, contraction and elimination are commutative and calls a minor of a (0, :kl) matrix any submatrix obtained by performing a sequence of the three operations. Unfortunately, the matrix M above is also minimally non-ideal with respect to this alternative definition of minor. It follows that we need a different concept of "minor" to overcome the above difficulty. To this purpose, we augment the list of minor-taking operations for (0, :kl) matrices by defining two more operations. Definition 8. Let A be a (0, ± 1 ) matrix and J a subset of N. Then (i) the semi-deletion of J in A is the submatrix A \ \ J obtained by removing all rows with a 1 in some column in J and removing all zero columns; (ii) the semi-contraction of J in A is the submatrix A / / J obtained by removing all rows with a - 1 in some column in J and removing all zero columns. The operations of deletion, contraction, semi-deletion and semi-contraction performed on A have a clear counterpart in the (0, 1)-extension D ( A ) . In fact, we have the following proposition whose easy proof is left to the reader.

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Proposition 9. Let J be a subset of N. We have that: D(A\J) = D ( A ) \ p ( J ) / r ( J ) , D(A/J) = D ( a ) / p ( J ) \ r ( J ) , D ( A \ \ J ) = D( A ) \p( J), D(A//J) =D(a)\r(J). We define a weak minor of A to be any submatrix of A obtained by performing a series of deletions, contractions, semi-deletions and semi-contractions. By the above proposition, if A ~ is a weak minor of A then D(A t) is a minor of D(A). It follows that the order in which the series of operations is performed does not affect the result. More specifically, letting NI, N2, N3, N4 be (possibly empty) subsets of N such that N1 NNi = {3 (i = 2 , 3 , 4 ) and N2 NNi -- 0 (i = 1,3,4), we have that the weak minor obtained by deleting N1, contracting N2, semi-deleting N3 and semi-contracting N4, in any order, coincides with the weak minor

a' = a\ul/N2\\N3//N4.

(20)

Of course a minor is a weak minor, while the converse does not hold in general. Moreover, the elimination of a column corresponds to performing both a semi-deletion and a semi-contraction. Hence a minor according to Guenin (obtained by deletions, contractions and eliminations) is also a weak minor obtained by choosing N3 = N4 in (20). Finally, we have that the concept of weak minor also generalizes the concept of set covering submatrix [4] as described at the beginning of Section 2. In fact, any set covering submatrix of A has the form (20) where N1UN2 = ~}, N3 nN4 = (~ and N3 UN4 is the index set of non-monotone columns. In particular, by switching the columns with index in N3, we get a (0, 1) matrix from A ~. We can now define the following concept of a minimally non-ideal matrix. Definition 10. A minimally non-ideal (0, d:l) matrix is a non-ideal matrix with the property that every proper weak minor is ideal. We will now derive a characterization of minimally non-ideal (0, ~:1) matrices in the spirit of Lehman's theorem. A consequence of our characterization will be that if A is a minimally non-ideal matrix then Q(A) has a unique fractional vertex. Observe that the class of minimally non-ideal (0, i 1) matrices includes all the (0, 1 ) minimally non-ideal matrices and their switchings. Another class of minimally non-ideal matrices is given by the minimally non-balanced matrices defined by Truemper [ 8 ]. A minimally non-balanced matrix is a (0, d: 1) square matrix with two non-zero entries per row and per column and with the property that the sum of the entries is not a multiple of 4. Balanced (0, i l ) matrices have been extensively studied in [ 1 ]. It is easy to construct minimally non-balanced (0, 4-1) matrices which are not switchings of (0, 1) matrices. Moreover, we can observe that A is

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minimally non-balanced if and only if D ( A ) is the node-edge incidence matrix of an odd cycle. Finally, for each integer s >~ 2, there exists a minimally non-ideal ( 0 , + 1 ) matrix which is neither a switching of a (0, 1) matrix nor minimally non-balanced. We will call such a matrix Js. It is obtained from the special matrix Js defined by Lehman, by simply turning the first component of the first row from 0 to - 1 .

1110

J3=

0 0

1 0

By analogy, we denote by J1 the (2 × 2) matrix with the first component of the first row equal to - 1 and the other entries equal to 1. Observe that D(Js) = Js+l. The following theorem and the subsequent corollary characterize minimally non-ideal (0, :k 1 ) matrices in terms of their (0, 1) counterparts and provide a description of their structure. In particular, we will show that a (0, + 1 ) minimally non-ideal matrix A which is not a switching of a (0, 1) matrix is either a switching of Js or contains a minimally non-balanced row submatrix R. In the latter case, every row of A not in R has at least three non-zero entries. Theorem 11. Let A be a (0, i l )

matrix. Then A is minimally non-ideal if and only if

D( A ) is minimally non-ideal. Proof. Let N denote the set of columns of A, with INI = n.

Claim 1. If D ( A ) is minimally non-ideal, then A is minimally non-ideal. Since D ( A ) is minimally non-ideal, it follows that every proper minor of D ( A ) is ideal and, hence, every proper weak minor of A is ideal. Consequently, to show that A is minimally non-ideal, it is enough to prove that A is non-ideal. If no column j E N exists such that both p ( j ) and r ( j ) belong to D ( A ) , we have that A is monotone, that is, a switching of a (0, 1) matrix (namely D ( A ) ) . In this case, obviously, A is non-ideal. If, on the other hand, there exists some j C N such that both p ( j ) and r ( j ) belong to D ( A ) , we have that D ( A ) contains a row with only two non-zero elements (namely those corresponding to p ( j ) and r ( j ) ) . It follows that p ( D ( A ) ) = 2. By Lehman's theorem, either D ( A ) is Jn-1 or D ( A ) contains as a row submatrix the node-edge incidence matrix R' of an odd cycle and all other rows have at least three non-zero elements. In the first case, A is Jn-2, which is non-ideal. In the second case, A has a row submatrix R with the property that D ( R ) = R ~ and R is minimally non-balanced. Since Q ( R ) has the (unique) vertex (½ . . . . . ½) which satisfies all the constraints associated with the other rows of A, it follows that A is again non-ideal. Assume now that A is minimally non-ideal but D ( A ) is not minimally non-ideal. Since A is non-ideal, we have that D ( A ) is non-ideal and there exists a proper minor B

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of D ( A ) which is minimally non-ideal. But then there exists a proper weak minor B ~ of A such that B = D ( B ~) and, by Claim 1, B ~ is non-ideal, contradicting the minimality of A. Hence, the theorem follows. [] From the above theorem the following corollary can be easily derived. Corollary 12. Let A be a (0, ± 1 ) matrix. Then A is minimally non-ideal if and only if A is a switching o f Jn-~ or a switching o f a minimally non-ideal (0, 1) matrix or A contains a row submatrix R which is minimally non-balanced and every row o f A not in R has at least three non-zero entries. The above corollary and Lehman's characterization of the unique fractional vertex of a minimally non-ideal (0, 1) matrix immediately imply the following corollary. Corollary 13. I f A is a minimally non-ideal matrix, then Q( A ) has a unique fractional vertex. In particular, there exists a switching A t o f A such that A t and the unique fractional vertex Yc o f Q( A ~) satisfy one o f the following: (i) A t =- Jn-I andY~l = ( n - 2 ) / ( n - 1), -~i = 1 / ( n - l ) , f o r i = 2 . . . . . n; (ii) A t =_ Jn-I a n d £ j = (n - 1)In, xi = 1In, f o r i = 2 . . . . . n; (iii) A t contains a minimally non-balanced row submatrix R, every row of A ~ not in R has at least three non-zero entries and xi = ½, f o r i = 1 . . . . . n; (iv) A I is a minimally non-ideal (0, 1) matrix different from Jn-I and Yci = 1 / p ( A I ) , f o r i = 1 . . . . . n. In the case of minimally non-ideal d-complete matrices, it is possible to prove a sharper version of Corollary 12. Corollary 14. A d-complete (0, ±1 ) matrix with n columns is minimally non-ideal if and only if it is a switching o f Jn-I or a switching of a minimally non-ideal (0, 1) matrix. Proof. Suppose that A is a minimally non-ideal matrix with n columns which is neither a switching of J~-I nor a switching of a minimally non-ideal (0, 1) matrix. It follows that P N R v~ ~, where P and R are the sets of positive and negative columns of A, respectively. Moreover, by Corollary 12, the matrix A contains a row submatrix R which is minimally non-balanced and every row of A not in R has at least three non-zero entries. It follows that D ( A ) contains a square row submatrix B = D ( R ) which is the incidence matrix of an odd cycle of length q >/3. Let {il . . . . . iq} be the columns of D ( A ) ordered in such a way that, for each pair of consecutive indices ih,ih+l (with iq+l =-- il), there exists a row in B with ones in columns ih, ih+l and zeroes elsewhere. Let j E P A R and assume, without loss of generality, that p ( j ) =- i2 and r ( j ) =_ i3. Suppose that q ~> 5. Since A is d-complete, we must have that D ( A ) contains a row with two ones in columns il,i4 and zeroes

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elsewhere, contradicting the assumption that every row not in B has cardinality at least 3. It follows that q = 3. Hence, D ( A ) =_ J2 and A is a switching of Jl, a contradiction. [] The description of minimally non-ideal matrices given by Corollary 12 does not provide a characterization of (0, :[:1 ) ideal matrices in terms of forbidden weak minors. In fact, while any non-ideal matrix contains a minimally non-ideal weak minor, the following example shows that also an ideal matrix can contain a (minimally) non-ideal weak minor (obtained, in this case, by contracting the first two columns and eliminating the fifth).

A=

1

0

0

0

1

1

1

1

-1

1

Nevertheless, since a (0, t l ) matrix A is ideal if and only if its disjoint completion A + is ideal, we can try to characterize non-ideal matrices in terms of forbidden weak minors of their disjoint completions. Indeed, thanks to Theorem 4, we can prove the following theorem. Theorem 15. A (0,-4-1) matrix A is ideal if and only if every weak minor of A + is ideal. Proof. The "if" part is obvious. Hence, assume that A is an ideal matrix. It follows, by Theorem 4, that D ( A +) is ideal. Let B be a weak minor of A + which is non-ideal. It follows that the extension of a fractional vertex of Q ( B ) is a fractional vertex of Q ( D ( B ) ) , and so D ( B ) is non-ideal. Since D ( B ) is a minor of D ( A +) and all the minors of D ( A +) are ideal, we have a contradiction. [] The above theorem implies that a ( 0 , + 1 ) matrix A is ideal if and only if its dcompletion A + does not contain a minimally non-ideal weak minor whose structure is described by Corollary 12. In the rest of this section, we will derive a sharper forbidden weak minor characterization of (0, ~:1) ideal d-complete matrices by using the result stated in Corollary 14. Namely, we will show that the only forbidden weak minors are the switchings of Js and of (0, 1) minimally non-ideal matrices. In proving the above result, we will also derive some crucial properties of logical and disjoint completions of (0, i l ) matrices which are interesting on their own. Observe first that Corollary 14 and Theorem 15 do not immediately imply the sharper forbidden weak minor characterization stated above. In fact, it is not always true that every weak minor of a d-complete matrix is d-complete. In other words, the property of being d-complete is not hereditary with respect to taking weak minors. Indeed, it can be easily checked that Js --- J+ but the proper (weak) minor of a+s obtained by contracting any column in the set {2 . . . . . s + 1} does not coincide with

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its disjoint completion. Moreover, the matrix is has also the property that its disjoint completion does not coincide with its logical completion. As a matter of fact, the following Theorem 16 shows that the coincidence o f the disjoint and the logical completion o f a (0, i l ) matrix A is a necessary and sufficient condition for all the weak minors of A + to be d-complete. Moreover, Theorem 17 will show that the matrices Js (s ~> 1) are (up to switching) the only matrices that one needs to forbid as weak minors of a d-complete matrix in order to guarantee the coincidence o f disjoint and logical completions and, hence, to turn d-completeness into a hereditary property. T h e o r e m 16. Let A be a ( 0 , + 1 ) matrix. Then A + ~ A ~ if and only if there exists a weak minor B o f A + satisfying B 4: B +. Proof. ( " i f " ) Let B be a weak minor o f A + such that B v~ B +. We have that there exist two rows o f B, say q and t, and a column j such that (PqURq)N(PtURt) = PtnRq = {j}, and with the property that the disjoint implication w o f q and t is not dominated by any row o f B. Let q/ and t I be the rows of A + whose restrictions to B are q and t, respectively. Since B is a minor o f A + we have that the signs of the entries of rows q~ and t ~ must agree in every column outside B. It follows that rows q~ and t t are the antecedents o f an implication w ~. Assume that A + = A% Hence, a row dominating w ~ is in A + and the restriction o f such a row to B is dominated by a row w H of B. But then w ~ dominates w, a contradiction. It follows that A + v~ A ~. ("only if") Assume now that A + :/: A% Hence, there exist two rows of A +, say q and t, and a column j such that Pt N Rq = {j}, Pq n R t = 9, ( Pq n Pt) U ( Rq N Rt) 4: 9, and with the property that the implication o f q and t is not dominated by any row in A +. Let B be the weak minor o f A + defined by B = A + / ( P q N Pt)\(Rq N gt). B contains the restrictions o f rows q and t, say q~ and t/, which are the antecedents of a disjoint implication. Moreover, B does not contain any row which dominates such a disjoint implication. In other words, B is different from B + and the theorem is proved. [] In the following we shall say that a matrix A is a minimal matrix with respect to a given property if A satisfies the property but none o f its (proper) weak minors does.

Let A be a (0,-4-1) d-complete matrix which is minimal with respect to the property that A 4: A °. Then A is a switching of is. T h e o r e m 17.

Proof. Since A :~ A °, there exist two columns in A, say j and k, and two rows, say q and t, such that P t n Rq = {j}, Pq N Rt = 9, k C (Pq N t~t) U (Rq N Rt), and with the property that the implication o f q and t is not dominated by any row in A. We can assume that the columns of A have been switched in such a way that Rq = { j } , Rt = and hence k c (Pq N Pt). Let N denote the set of columns of A.

Claim 1. Pq U Pt = N. In fact, assume that there exists a column i ~ PqUPt (observe that i must be different from j and k). Hence, the weak minor B = A - {i} contains the restriction of rows

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q and t, but does not contain any row which dominates their implication. This is a contradiction to the hypothesis that B satisfies B + = B%

Claim 2. For any column l E N - {j, k}, there exists a row r I o f A whose restriction to A / { l } dominates the restriction o f either row q or row t and such that I E Pr, In fact, the weak minor B = A / { l } satisfies B + = B ~. Hence, we have that there exists a row rl of A whose restriction to B dominates the restriction of one of the two rows. Evidently, since row rt dominates neither row q nor row t, we have that row rl has a 1 in column I. Claim 3. For any column l E N - {j, k}, we have that row rl has a non-zero in column j, a 0 in column k and non-negative values in all the other entries. Assume first that the restriction of row rt to B = A / { l } dominates the restriction of row q. As a consequence, row q has a 0 in column I. Moreover, since Rq = {j}, we have that the only possible negative entry of row rt is in column j. It follows that row rl has a non-zero in column j, otherwise rt would dominate the implication of q and t, a contradiction. In particular, rl has a - 1 in column j, since its restriction to B dominates the restriction of q. Finally, assume that row rs has a 1 in column k. Since Pq U Pt = N and 1 ¢ Pq, we have 1 E Pt. It follows that no row in B can dominate the restriction of row t, and hence the restrictions to B of rows t and rt form the antecedents of a non-disjoint implication. Since B + = B ~, we have that there exists a row r / in A whose restriction to B dominates such an implication. But then row r ~ dominates the implication of rows q and t in A, a contradiction. It follows that row rl has a 0 in column k. An analogous argument can be used to prove the claim in the case in which the restriction of row rt to B dominates the restriction of row t. Claim 4. There do not exist rows rt and rm with the property that j C Pr~ N Rr,,,. Suppose conversely that such rows exist. Consider the weak minor B = A - {k} and observe that every row of B + is the restriction of a corresponding row of A. Since, by Claim 3, both rows rl and rn, have a 0 in column k, we have that their restrictions belong to B and are the antecedents of an implication w. Moreover, since B + = B *, we have that w is dominated by a row w ~ of B +. But then w~ is the restriction to B + of a row of A which dominates the implication of rows q and t, a contradiction. It follows that we can assume without loss of generality that column j has been switched (possibly interchanging rows q and t) in such a way that j E Pr~ for any 1 c N-

{j, k}. Hence we have the following.

Claim 5. All the entries o f row rl (l C N - {j, k}) are non-negative. Claim 6. For any 1 C N - {j, k}, the restriction o f row rl to A / { l } dominates the restriction o f row t. In fact, observe that the restriction of row rl t o the minor A / { l } cannot dominate the restriction of row q (since row q has a - 1 in column j ) and hence, by Claim 2, it must dominate the restriction of row t.

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Claim 7. Row t has a 1 in columns j and k and 0 elsewhere. Assume, conversely, that row t has a non-zero entry in a column l ~ N - {j, k}. By assumption, such an entry is a 1. Since row rt does not dominate row t, we have that there exists a column i E N - {j, k, l} with a 1 in row rl and 0 in row t. But then the restriction of rl to A / { 1 } does not dominate the restriction of t, contradicting Claim 6. Claim 8. For any 1 E N - {j, k}, row rt has a 1 in columns j and l and 0 elsewhere. In fact, by Claim 7, this is the only possibility to ensure that the restriction of rt to A / { l } dominates the restriction of t. Observe that, by Claims 1 and 7, we have that Pq = N - {j}. It follows, also using Claim 5, that rows q,t, rl (l E N - {j,k}) form the matrix a~ (s = IN I - 1). To conclude the proof we have to show that no other row exists in A. Assume conversely that there exists in A a row r' ~ {q, t} U {rt: l E N - {j, k}}. We have that there exists a column i ~ j such that i ERr,. In fact, otherwise, either row r ~ dominates row q ( i f j E R r , ) or it dominates row t (if Pr' = {j}) or it is dominated by some row in {t} U {rt: l E N - {j, k}}. In any case we get a contradiction. It follows that the weak minor B / / { i } is a proper minor of A with the property that B + ¢ B*, a contradiction. Hence, the theorem follows. [] Corollary 18. Let A be a (0, :t:l) d-complete matrix. If A does not contain a switching o f is as a weak minor then every weak minor B of A is d-complete.

Proof. By Theorem 17, if A does not contain a switching of ,Is as a weak minor then A ~ A% Hence, by Theorem 16, every weak minor B of A is d-complete.

[]

As a consequence of the above result, Corollary 14 and Theorem 15 immediately imply the following characterization of ideal matrices in terms of forbidden weak minors of their disjoint completions. Corollary 19. Let A be a (0,4-1) matrix. Then A is ideal if and only if A + does not contain, as a weak minor, a switching o f Js or a switching o f a minimally non-ideal (0, 1 ) matrix. If A =- A* then we have the following interesting corollary. Corollary 20. If A =~ A ~, then A is ideal if and only if A does not contain, as a weak minor, a switching o f a minimally non-ideal (0, 1) matrix. Proof. Since A ~ A ~' implies A -- A +, we have simply to show, by Corollary 19, that A does not contain as a weak minor a switching of Js. Assume, conversely, that a weak minor B of A is a switching of Js and denote by S the columns of A not in B. Let t and q be the rows of A whose restrictions are the first two rows of B. Evidently, no column in S has entries of opposite sign in rows t and q. It follows that t and q are the antecedents of a logical implication w in A. Since A ~ A ~', A contains a row w t which

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d o m i n a t e s w. But, since B contains the restrictions o f rows t and q, it also contains a row which d o m i n a t e s the restriction o f w'. Such a row dominates also the first row o f B, a contradiction.

[]

O b s e r v e that H o o k e r ' s theorem as stated in the previous section is an i m m e d i a t e c o n s e q u e n c e o f the above corollary.

Acknowledgements W e wish to thank Bertrand G u e n i n w h o p r o v i d e d a better statement and a shorter p r o o f for T h e o r e m 17 and w h o s e suggestions and c o m m e n t s were invaluable for i m p r o v i n g the presentation o f this paper.

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