Fourth IFIP International Conference on Theoretical Computer Science - TCS 2006: IFIP 19th World Computer Congress, TC-1, Foundations of Computer Science, ... in Information and Communication Technology)

*,M= E

"^1' "X'-1^1" (/(.)-/(5-M)), '

SCN, xes

(1)

'•

where l^l and |A''| denote the size of S and C, respectively, or alternatively by 1 •^Z^^) = IM E ifirr^i^'^) U W) - fimiT^,^))),

(2)

TTGil/V

where 11N is the set of all one to one mappings (enumerations) w : N {!,..., |iV|} and m('K,x) = {y E N \ ^(y) < 7r(x)} is the set of all y G A^ arranged on the left side of x.

3 A vertex rating for directed graphs We now define a characteristic function fa to rate the vertices in directed graphs. The rating will measure the influence of a vertex on the connectivity

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structure. The smaller the rating of a vertex the greater its importance to the connectivity. Let G = {VG,EG) be a finite directed graph, where VG is a finite set of vertices and EG C VG X VG is a finite set of directed edges. A path in G is a sequence p = {vi,... ,Vk), k > 1, oi distinct vertices such that (vi,Vi+i) G EG for i = 1 , . . . , A; — 1. We say, p is a path of length k from vi to Vk- A path is called a cycle of G if G additionally has edge {vk,vi). We will consider only simple paths and cycles in which all vertices are distinct. For a vertex set V C VQ, let G\v' be the subgraph of G induced by the vertices o f y , that is, G|K' = (VQ', Ec) where VQ' =V' undEo' =-EnV^'xV^'. G is strongly connected if for every pair of vertices u,v G VG there is a path from u to u in G. A strongly connected component of G is a maximal strongly connected subgraph of G. Let SCC(G) be the set of all strongly connected components of G, and / G be a function from the subsets of VG to the real numbers R (here we need only the set of non-negative integers) such that for every subset V C VG, / G ( n = |SCC(G|vOIThat is, / G ( ^ ' ) is the number of strongly connected components in the subgraph of G induced by the vertices of V. Note that / G is a characteristic function, because / G ( 0 ) is always zero. The complete vertex set VG is always the only carrier of / G for every directed graph G. By Axiom 2, we have

J2haiv)=fG{VG)

= \SCG{G)\.

veVa

Figure 1 shows an example of the vertex rating (j)f^ for a directed graph G with vertex set VG = {f i, ^'2, ^3, V4, v^, VQ, v-r, v^}. Since G is strongly connected, we get fciVo) = Y^veVa'^foi'") = 1- Following the computation of (j)fc by Equation 2, vertex vs has rating (j)f^{v8) = 5, because /G("^(7'',U8) U {VS}) — /G(m(7r, Vg)) = 0 if and only if 7r(v6) < 7r(z;8). Otherwise, we have fG{m{TT, vs)D {t^s}) —/G(TO('?r, Vg)) = 1, which happens for half of all 8! enumerations n. Vertex vi has rating (pfa{vi) = | , because fG{'m{Tr,vi) U {vi}) — /GC^^C""",fi)) = 0 if and only if 7r(f2) < 7r(vi) and 7r(i;3) < 7r(wi). Otherwise, we have /G(m(7r, fi) U {vi}) — fGi'm{n,vi)) = 1. Here the second case happens for two third of all 8! enumerations TT. Let G = {VG,EG) and G' = {VG',EG') be two directed graphs. We call G and G' isomorphic if there is a one to one mapping 6 : Vcj —» VG' such that for every pair of vertices z^i, 112 € VG, {vi,v2)e

EG <S=> {b{vi),b{v2))

eEG'.

Such a mapping b is called an isomorphism between G and G'. If G and G' are isomorphic then / G ( ^ ' ) = fG'{b{V')) for every vertex set V C VG- Here

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Fig. 1. The vertex rating (j>f^ for a directed graph G with 8 vertices. The smaller the rating of a vertex the greater its importance to the connectivity of the graph. h{y') = {h{u) I u G V'} is the image of V under b. This implies 4'fai''^) = 4'ja' (^(^)) f^'^ ^"^ vertices v € V. Let V C VQ be any set of vertices of G. Graph G is called V-symmetric if for every pair of vertices vi,V'2 S V there is an isomorphism 6 of G to G itself such that b{yi) = V2- In ^'-symmetric graphs all vertices v E V have the same rating. If two vertices Vi, V2 have the same neighborhood, i.e., if {u \ {u,vi) G EQ} = {u | {u,V2) € EG} and {u I {vi,u) G EG} = {u I (i>2, u) G EG}, then G obviously is {wi, wgj-symmetric. Figure 2 shows some examples of partially symmetric graphs.

Fig. 2. The graph to the left is {i)i,'i;3,t;5,ii7}-symmetric and {v2,V4,ve,vs}symmetric, the graph in the middle is {iii,i)2,t'3}-symmetric, and the graph to the right is {iiiji^aji'a, V4, «5, V6}-symmetric. The computation of a vertex rating 0/^ {v) by Equation 1 or Equation 2 is highly inefficient. The number of subsets and the number of enumerations increase exponentially in the number of vertices of G. To handle the computation of (j)f^ for many practically interesting instances we will introduce a method to decompose a large graph into smaller parts. This decomposition will allow us to compute efficiently the ratings of vertices of the original graph by using the ratings of the vertices of smaller subgraphs. Our decomposition method will be introduced by the following two lemmas and Theorem 1. The first lemma shows that the computation of a rating 4>fa{v) for which the arguments of fa are restricted to vertices of a subset V C VG yields the computation of 0/^. ^ (v).

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Lemma 1. Let G = {VG,EG) be a graph, V C VQ, and G' = G\v'- Let Ilva be the set of all enumerations n : VG —»• { 1 , . . . , | V G | } . Then for every vertex veV ha' (^) = riTTl E

(fG(.{rn{n, v) U {v}) n V) - /G(m(7r, v) D V')).

Proof. Let Lfv be the set of all enumerations TT' : V —> { 1 , . . . , | y |}. First we show that for every enumeration TT' e Uv there are (|V''| + l ) - ( | y | + 2 ) \VG\ unique enumerations 7r e LIVG ^^ch that for every pair of vertices vi,f2 € V, n'{vi) < TT'{V2) if and only if 7r(ui) < IT{V2). Let p = {vi^,.. .,Vi^^,^) be the sequence of vertices of V in the order defined by TT', that is n'ivi,)


If we consider the vertices of VG — V in an arbitrary order, then the first vertex of VG — V can be placed at | y | + 1 positions at sequence p to get a sequence with \V'\ + 1 vertices. After that the next vertex can be placed at | y | + 2 positions in the resulting sequence to get a sequence with | y | + 2 vertices, and so on. The final vertex of VG — V can be placed at | VQ | positions in the sequence obtained by the preceding placement to get a sequence of all |VG| vertices of G. For all these ( | y I + 1) • (|T^'| + 2) |VG| enumerations TT defined for enumeration TT' we have fG'im{TT',v) U {v}) fG'{m{n',v)) = / G ' ((m(7r, v) U {t;}) n V) - fG' (m(7r, v) n V) for every vertex v £V', and thus -^/o'W = WvX.'en^XfG'{m(.ir',v)VJ{v})-fG'{m{'K',v))) _ _1_Y- \V'\\ ^^enva

= ^

(/»/ ((m(7r,i;)U{-»})nV')-/„, {m{-K,v)nV')) {\V'\+l)-{\V'\+2) \VG\

E^envG ifaiimin,

v) U W ) n ^ ) - fcimi-,,

v) n F'))-

The last equaUty follows from the fact that fG'{V" n V) = fG{V" n V) for every subset V" CVQ• It is easy to see that the rating of a vertex in a graph G depends only on the connectivity structure of the strongly connected component the vertex belongs to, as the following observation shows. If G' = {VG',EG') is a strongly connected component of G = (VG, EG) then for every vertex v G VG' and every vertex set V" C VG, faiiV" U {v}) n VG') - faiV" n VG') - fG{V" U W ) and thus by Lemma 1, (pj^, (f) = ^/(,(v).

fG{V"),

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We will now define a property of a vertex pair u, v that allows us to compute independently the rating for two vertices u and v. That is, the rating of M in G will be equal to the rating of u in graph G without v. Definition 1. Let G = (VQ, EQ) he a directed graph and u,v £VG be two nonadjacent vertices, that is, neither {u,v) nor {v,u) is an edge ofG. Vertex u and vertex v are strongly separable in G if for every strongly connected induced subgraph H = {VH,EH) of G which contains u and v there is a strongly connected subgraph J = {Vj,Ej) of H without u and v such that H\VH-VJ has no path from u to V and no path from v to u. For the proof of the next lemma we need the notion of an undirected graph. In an undirected graph G = iVc, EG) the edge set is a subset of {{u, v} \u,v & VG, U ^ v}. Analogously to the definitions for directed graphs, an undirected path of length A;, fc > 1, is a sequence p = (vi,... ,Vk) of fc distinct vertices such that {vi, Vi+i} S EG iov i = 1,... ,k — 1. An undirected path is called an undirected cycle if G additionally has edge {vk,vi} and the path has at least three vertices. The subgraph of G induced by a vertex set V Q VG has edge set EG r\{{u,v} \ U,V G V, U y^ V}. A graph is connected if there is a path between every pair of vertices, a connected component is a maximal connected subgraph, a forest is an undirected graph without cycles, and a tree is a connected forest. L e m m a 2. Let G = {VG,EG) be a directed graph and VH,VJ Q VG be two vertex sets such that VH U Vj = VG and for every edge (^1,112) G EG both vertices are in VH or in Vj, or in both sets. Let H = G\VH, J = G\vj, and I = G|y„nVj • V every pair of vertices u G VH — Vj, v £ Vj — VH is strongly separable in G, then for every vertex set V C VG, foiV)

= fniV

n VH) + fj{V' n Vj) - fj{V' n Vi).

Proof. Let V C VG be any set of vertices of G. Consider the following undirected graph T — (Vr, ET) with vertex set VT == SCC{H\v') U SCC(J|yO such that two vertices of VT are connected by an undirected edge if and only if the two strongly connected components have at least one common vertex. If two distinct strongly connected components of VT are connected by an undirected edge in T then one of them has to be from SCC{H\v') and the other has to be from S C C ( J | K ' ) - Furthermore, for every strongly connected component C of SCC{I\v'), there is exactly one strongly connected component Ci of SCC{H\v') and exactly one strongly connected component C2 of S C C ( J | v ) , and the common vertices of Ci and C2 are exactly the vertices of C. Since every pair of vertices u GVH — Vj, v €Vj — VH ^s strongly separable in G, the undirected graph T has no cycles, that is, T is a forest. The number of connected components of T (the number of trees of forest T) is equivalent to the number of strongly connected components of G. The number of connected

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components in a forest is always equivalent to its number of vertices minus its the number of edges. Since T has exactly one edge for every strongly connected component of S C C ( / | v ) and exactly one vertex for every strongly connected component of SCC(iJ|y') and S C C ( J | y ) , we get /G(V')

= fniV n VH) + fj{v' n Vj) - fiiV n Vj). a

The following theorem states how ratings of vertices of G can be computed by the ratings of the same vertices in certain subgraphs of G. Theorem 1. Let G = (VCEG) be a directed graph and VH,VJ Q VQ be two vertex sets such that VH U Vj = VQ and for every edge (vi,'y2) € EQ both vertices Vi,V2 are in VH or in Vj, or in both sets. Let H — G\VH, J = G\vj, and I = GlvnnVj • If every pair of vertices u GVH — VJ, V € VJ — VH is strongly separable in G, then 1. for every vertex w GVH (^ Vj, 4>f^ {w) = (f)f^ (w) + (pfj {w) — 4>fi (w), 2. for every vertex w &VH — Vj, 4>fG (^) — 4'fH (^)» ^''^'^ 3. for every vertex w GVJ — VH, (/>/G(W) = (f)fj{w). Proof. Let w be any vertex of VQ- By Lemma 2, for every vertex set V fciV

U {w}) - faiV)

=

ifHiiV

CVG,

U M ) n VH) - fniV' n VH))

+ {fj{{v'yj{w])nvj) -{fi{{V'yj{w))nVi)

-fj{V'nVj)) -fj{v'nVi)).

If w e VH n Vj, then by Lemma 1 we get 4>ia (w) =
If w is a vertex of VH-VJ, V r\Vi, and thus fciV

{w).

then (V' U {w}) nVj = V'nVj and {V U{w})nVi

U {w}) - faiV)

=

=

fH{{V'U{w})nVH)-fH{V'nVH),

which implies by Lemma 1 f„{w). If w is a vertex of Vj — VH, then an analog argumentation yields cpf^lw) =
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Fig. 3. Four graphs G = (VG,-EG), H = G\v„, J = G\vj, and I = G\v„nVj such that VH,VJ C VG, VH U VJ = VQ, and for every edge (vi, V2) £ EG both vertices are in VH or in Va, or in both sets. Vertex pair vi & VH — Vj, ve G Vj — VH is strongly separable in G. polynomial time algorithms which decides whether two vertices in a directed graph are not strongly separable, unless P = NP. The NP-hardness follows by a simple reduction from the satisfiability problem. The terms we use in describing this problem are the following. Let X = {xi,..., x„} be a set of Boolean variables. A truth assignment for X is a function t : X —^ {true, false}. If t{xi) = true we say variable Xj is true under t; if t{xi) = false we say variable Xi is false under t. If Xi IS Si variable of X, then x, and xj are literals over X. Literal Xi is true under t if and only if variable Xj is true under t; literal xj is true under t if and only if variable Xi is false under t. A clause over X is a set of literals over X, for example {xi,xj, X4}. It represents the disjunction of literals which is satisfiedhy a truth assignment t if and only if at least one of its literals is true under t. A collection C of clauses over X is satisfiable if and only if there is a truth assignment t that simultaneously satisfies all clauses of C. The satisfiability problem, denoted by SAT, is specified as follows. Given a set X of variables and a collection C of clauses over X. Is there a satisfying truth assignment for C? This problem is NP-complete even for the case that every clause of C has exactly three distinct literals (3-SAT, for short). T h e o r e m 2. The problem to decide whether two vertices u,v of a directed graph G are not strongly separable is NP-complete. Proof. Let us first illustrate that the problem belongs to NP. Two vertices u and V are not strongly separable in G if and only if G has a strongly connected induced subgraph G' = (VG',EG') that includes u and v such that G"|VQ,_{„_„} has no strongly connected subgraph G" = (VG",EG") such that in G'\v^,-Va>i there is no path from u to v and no path from v to u. Without loss of generality we can assume that G" is a strongly connected component of G' | y^, _ |„_^,}. So we can non-deterministically consider every strongly connected subgraph G' of G that includes u and v. Then we can verify in polynomial time for every strongly connected component G" — {VG",EG") of G'\v^,-^u,v} whether G'\v^,-Va'' has no path from u to v and no path from v to u. Thus, the problem to decide whether two vertices u, v are not strongly separable belongs to NP.

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The NP-hardness follows by a simple transformation from 3-SAT. Let X = { x i , . . . , Xn} be a set of n Boolean variables and C = { C i , . . . , Cm} be a collection of m clauses. We define a graph G{X, C) with two vertices u, v such that there is a truth assignment t for X that satisfies every clause of C if and only if u and v are not strongly separable in G{X,C). Figure 4 shows an example of such a construction for four variables a;i,a;2,X3,X4 and four clauses { X 2 , X 3 , X i } , {xi,X^,X4},

{xl,X^,X^},

{xT,X2,Xs}.

Graph G{X,C) has six vertices u,a,b,v,c,d, two literal vertices Xj, x7 for every variable Xi, 1 < i < n, and three literal vertices Cj^i, Cj,2, Cj,3 for every clause Cj = {cj,i,Cj,2,Cj,3}, I < j < rn. G{X,C) has the edges {u,a), (a,xi), (a,xl), the edges (xi,Xi+i), ( x i , x ^ ) , (x7,Xi+i), ( x 7 , x ^ ) for i = 1 , . . . , n 1, the edges (x„,6), (x^^, &), {b,v), {v,c), (c,ci,i), (c,ci,2), (c,ci,3), the edges {cj^k,Cj+i,i) for j = l , . . . , m — 1 and k,l G {1,2,3}, and the edges {cm,i,d), {cm,2,d), {cm,3,d), {d,u), and {d,a). Additionally, there are a so-called cross edges from every literal vertex Xj (x7) for variable Xi to every literal vertex xj (xj, respectively) for some clauses. In Figure 4, the cross edges are drawn as dotted arcs.

literal vertices for variables

literal vertices for clauses

Fig. 4. The graph G{X,C) for X = a;i,a;2,a;3,a:4 and C = {a;2,a;3,a;4}, {xi,0:2,2:4},

Every cycle of G{X, C) that includes vertex u and v consists of two vertex disjoint path pi = {u,a,..., b, v) and p2 = {v,c,..., d, u). Path pi passes exactly one literal vertex for every variable, and defines in this way an assignment t for the variables, where path p2 passes exactly one literal vertex for every clause. Assume there is a truth assignment t for X that satisfies every clause. Then there is an induced subgraph G' of G{X, C) that includes vertex u and v but no cross edge, for example the subgraph of G{X, C) induced by u, v, a, b, c, d and all true literal vertices. In this case, it is not possible to destroy all paths between u and V and all paths between v and u by removing a strongly connected subgraph of G'. Thus u and v are not strongly separable. Assume there is no truth assignment for X that satisfies every clause. Then every strongly connected induced subgraph G' of G that includes u and v has

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at least one cross edge («', v'). In this case it is easy to destroy all paths from u to V and all path from v to M by removing a cycle that includes the edge (d, a) and the cross edge {u',v'). Thus u and v are strongly separable. D Theorem 2 can be used to prove that deciding whether two vertices have a different rating is NP-hard. Consider again the graph G{X, C) with the two vertices u and v constructed for an instance {X, C) of 3-SAT as in the proof of Theorem 2. Let G'{X,C) be the graph G{X,C) without the vertex v and its incident edges. Then 4'fo(x,c) (") ~ ^fa'tx o (^) ^^ ^ ^^^ ^ ^^^ strongly separable in G, and (l>fg,x c) (•") < 'PSG'IX O (") '^^ " ^'^'^ ^ ^^^ '^°^ strongly separable in G. Theorem 3. The problem to decide whether (pf^ (u) < (pf^ {v) for two vertices u,v of a directed graph G is NP-hard. Thus, an algorithm for the computation of 4>f^ can be used to decide an NP-hard as well as a co-NP-hard decision problem.

4 A vertex rating for undirected graphs The vertex rating (j)f^ for directed graphs can simply be extended to undirected graphs. For an undirected graph G let dir(G) be the directed graph we get if we replace every undirected edge {u,v} by two directed edges {u,v) and {v,u). Let fa now be the function from the subsets of VQ to the real numbers R such that for every V CVG, / G ( ^ ' ) is the number of connected components in the subgraph of G induced by the vertices of V. That is, the rating of a vertex v in an undirected graph G is equal to the rating of v in the directed graph dir(G). Figure 5 shows an example of the vertex rating cpf^ for an undirected graph G with vertex set VQ = {VI,V2,V3,V4,V5,VQ,V7,V8}.

Fig. 5. The vertex rating (j)f^ for an undirected graph G with 8 vertices. It is easy to verify that two vertices u, v of dir(G) are not strongly separable if and only if G has a chordless cycle that includes u and v. A chord for a cycle c = ( u i , . . . ,Uk) is an edge {ui,Uj} such that 2 < |i — j | < k — 2. The problem of determining whether an undirected graph G contains a chordless cycle can be solved in linear time [3, 15, 17]. This is the well-known chordal

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graph recognition problem. A graph G is a chordal graph if any cycle of G of length at least four has at least one chord, or alternatively, if G has no chordless cycle, see [8]. The problem of determining whether G contains a chordless cycle of length fc > 5 can be solved in 0 ( | V G | + |-EG|^) time on 0 ( | V G | • \EG\) space, see [11]. Theorem 1 applied to undirected graphs yields the following theorem which is a more general version of Proposition 2 of [7]. Theorem 4. Let G — (VcEa) be an undirected graph and VH,VJ C VG he two vertex sets such that V/f U Vj = VG and for every edge {vi,U2} G EG both vertices ui,U2 are in VH or in Vj, or in both sets. Let H = G\VH, J = G\vj, and I = G\vHnVj- If G has no chordless cycle with a vertex ofVn — Vj and a vertex of Vj — VH, then 1. for every vertex w £ VH r\Vj, 0/^ (w) = (pf„ (w) + (j)fj (w) — (j)fj (w), 2. for every vertex w GVH — Vj, 1 connected components Gi,... ,Gk by removing a complete subgraph / of G. Let G'^ = G\va-uVi for i = 1 , . . . , fc be the subgraphs of G induced by the vertices of connected component Gi and the vertices of the removed complete subgraph / . Then by Theorem 4 for every i = 1 , . . . , fc the vertex rating for a vertex w of G'^ is ^fo (•"') = ^fc'. ("') ^^^ ^^^ vertex rating for a vertex iz; of J is k

An example of a class of graphs for which the rating (j>f^ is efficiently computable is the class of cycle composed graphs which can recursively be defined as follows. The cycle C„ with n > 3 vertices is cycle composed. Let G = (VG, EG) be a cycle composed graph and ei = {ui, ui} be an edge of G. Let C„ = {Vc„,Ecn) be a cycle with n> 3 vertices and 62 = {w2, ^"2} be edge of Cn- Then the graph obtained by the vertex disjoint union of G and C„ and the identification of U2 with Ml and V2 with vi is cycle composed. That is, the composed graph has vertex set VcjUVcn -{u2,V2} and edge set {{h{u),h{v)} \ {u,v} € EQUEC^}) where h{u) — u for every M G VG U Vc„ — {^2,^2}, and h{u2) = MI and h{v2) = vi. Cycle composed graphs are biconnected and have tree-width at most 2, see [13, 1] for a definition of tree-width. Graphs of tree-width at most 2 can be recognized in linear time by removing vertices of degree at most 2. When a vertex u of degree 2 is removed then the two neighbors of u will be connected by an edge if they are not adjacent. A graph has tree-width 2 if and only if it can completely be reduced by removing vertices of degree at most 2 in the way described above, see for example [19]. Let C be the set of vertex sets of the cycles used to compose a cycle composed graph G, that is, C has a vertex set C for every cycle used to compose G. The

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vertex rating 0(w) for all vertices w of G is computable in linear time by the following simple procedure. 1. for every u E V do { 2. let<^/^(u):=(deg(u)-2)*(-i);} 3. for every C £ C do { 4. for every u £ C do { 5. let (l)f^(u) := (pfa(M) + 1^; } } Since a vertex u is involved in degr(u) — 2 vertex identifications, the rating for u can be initialized by (deg('u) — 2) * (—|). After that the algorithm adds for every cycle C the fraction j ^ to the ration of every vertex of C. Since the number of vertices in the sets of C is ^cec 1^1 ~ 2|£^G| — \^G\, the rating for all vertices in cycle composed graphs can be computed in linear time, if C is given. The vertex sets of the cycles can be computed by the following algorithm. We assume that an empty vertex list is initially assigned to every edge. That is, every edge {u,v} is initially represented as a pair ({u,u},0). An edge e = {{u,v},L) with a non-empty vertex list L represents a path between u and v passing the vertices of L. If G has a vertex u of degree 2 such that the two neighbors v,w of M are not adjacent, we remove vertex u and its two incident edges {{u,v},Li), {{u,w},L2) and insert a new edge {{v,w},Li U L2 U {u}) between u and v. If G has a vertex u of degree 2 such that the two neighbors VjW of u are adjacent, the vertices of a cycle can be reported. Let {{u,v},Li), {{u,w},L2)-, {{v, w}, L3) be the three edges between the vertices u, v and w. The algorithm then reports vertex set Li U L2 U L3 U {u,v,w}. If graph G has no further edges than the three edges above, then all cycles are reported and the algorithm finishes. If graph G has some further edges and L3 is non-empty, then the graph is not cycle composed. In any other case the algorithm removes the two edges {{u,v},Li), {{u,w},L2) and so forth. If this processing ends because there are no further vertices of degree 2, then the graph is also not cycle composed. This algorithm computes the vertex sets of all cycles used to compose a cycle composed graph. The running time of this algorithm is 0 ( | V G P ) because we have to check for every vertex whether its two neighbors are adjacent. However, this problem can be eliminated by a simple trick which is also used in [19] for the recognition of outerplanar graphs. The trick is to check whether the two neighbors v, w of u are adjacent at the time when one of these two vertices V, w gets a degree of 2 or less. At that point the test can be done in a fixed number of steps and either a new edge is inserted or a cycle is reported. This modification yields a linear time algorithm for the computation of all cycles of a cycle composed graph. The following example shows a possible implementation.

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create-new-edge (vertex u) { let ei = {{u,v},Li),e2 = {{u,w},L2) € EG be the two edges incident to u; insert {{v, w}, Li U -L2 U {u}) into Enew', remove ei and 62 from EG; if (deg^^ (u) = 2) then insert i; into M; if (deg£;^(w) = 2) then insert w into M; } raove-new-edge (edge enew = {{u,v},Lnew)) { if there is an edge e = {{u, v}, L) G EG then { o u t p u t L U Lnew U {u, v]]

remove Cnew from Enewl if ( | £ G | = 1) and (l^newl = 0) then halt " all cycles reported"; else if (L ^ 0) then halt " G is not cycle composed"; else { remove enew from -Enew; insert Cnew into EG] if (deg^;^ (u) = 3) then remove u from M; if {deg^^iy) = 3) then remove v from M; }

} compute-cycles (graph G = {VG,EG)) { let M := 0; for every u GVQ do { a (degE^iu) = 2) then { insert u into M; } } while (M 5^ 0) { let u€ M; if there is an edge enew S -Enew incident to u then move-new-edge (e else { if (deg£;^(u) = 2 ) then create-new-edge (u); remove u from M; } } halt " G is not cycle composed"; }

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The algorithm above stores in a set M all vertices of degree 2. Note that the degree of a vertex is always determined by the edges of EQ • For every vertex u adjacent with exactly two vertices v,w a. new edge is inserted into a set denoted by -Bnew but not yet into edge set EG of graph G. Whenever a vertex w of M is considered for processing it is first checked whether there are edges incident to u in set -Bnew If -E'new has an edge e incident to u then e will either be inserted into EG (if the two vertices of e are not adjacent by some edge of EG), or a cycle is reported (if the two vertices of e are adjacent by some edge of EQ)- The test whether the two vertices of e are adjacent by some edge of EG can be done in time 0(1) because u is one of the end vertices of e and has vertex degree 2. This proves the following theorem. Theorem 5. The vertex rating 4>f^{u) for all vertices u of a cycle composed graph G is computable in linear tim,e. The vertex rating (l>f^ is also computable in linear time for chordal graphs. An interesting characterization of chordal graphs is the existence of a perfect elimination order. Let p = ( u i , . . . , u„) be an order of the \VG\ = n vertices of G = {VG,EG), and let N{G,p,i) for i = 1 , . . . ,n be the set of neighbors Uj of vertex Ui with i < j , N{G,p,i)

:= {uj I {ui,Uj} € EG A i < j}.

The vertex order p — (ui,..., w„) is called a perfect elimination order (PEO) if the vertices of N{G,p, z) for i = 1 , . . . , n — 1 induce a complete subgraph of G. Dirac [3], Fulkerson and Gross [5], and Rose [14] have shown that a graph G is chordal if and only if it has a perfect elimination order. Rose, Tarjan, and Lueker have shown in [15], that a perfect elimination order can be found in linear time if one exists. If a perfect elimination order p = {vi,..., Vn) of the vertices of G = (VQ, EQ) is given, then the vertex rating (j)f^ can be computed with Theorem 4 by the following algorithm. Note that, in a complete graph G with n vertices, I/I/Q (V) = - for every vertex of G, because G is Vc-symmetric. 1. let (pfoivn) ••= 1;

2. for i = n — 1 , . . . , 1 do { 3.

l e t (Pfa{Vi)

4.

for a l i v e A/'(G,p,j) do {

: = |;v(G,p,i)|+i'

5.

let (t>Sc{v) ••= 'Pfaiv)

+ \N{G,l,i)\ + l -

\NiG,p,i)V

>>

The running time of this algorithm is linear in the size of G, because the assignment of Line 3 is done exactly | VG | — 1 times and the assignment of line 5 is done exactly |£^G| times. Since the perfect elimination order can be found in linear time, we get the following theorem. Theorem 6. The vertex rating 4>fah^) f°''~ ^^^ vertices v of a chordal graph G is computable in linear time.

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References 1. H.L. Bodlaender. A partial fe-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209:1-45, 1998. 2. X. Deng and C.H. Papadimitriou. On the complexity of cooperative solution concepts. Methods of Operations Research, 19(2):257-266, 1994. 3. G. Dirac. On rigid circuit graphs. Ahh. Math. Sem. Univ. Hamburg, 25:71-76, 1961. 4. D.J. Felleman and D.C. Van Essen. Distributed hierarchical processing in the primate cerebral cortex. Cerebral Cortex, 1:1-47, 1991. 5. D.R. Fulkerson and O.A. Gross. Incidence matrices and interval graphs. Pacific J. Math., 15:835-855, 1965. 6. M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, San Francisco, 1979. 7. D. Gomez, E. Gonzalez-Arangiiena, C. Manuel, G. Owen, M. del Pozo, and J. Tejada. Splitting graphs when calculating Myerson value for pure overhead games. Mathematical Methods of Operations Research, 59:479-489, 2004. 8. A. Hajnal and J. Suranyi. Uber die Auflosung von Graphen in vollstandige Teilgraphen. Ann. Univ. Sci. Budapest, Eotvos Sect. Math., 1:113-121, 1958. 9. R. Kotter and E. Wanke. Mapping brains without coordinates. Philosophical Transactions of the Royal Society London, Biological Sciences, 360(1456) :751766, 2000. 10. R.B. Myerson. Graphs an cooperations in games. Methods of Operations Research, 2:255-229, 1977. 11. S.D: Nikolopoulos and L. Palios. Hole and antihole detection in graphs. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 850-859. ACM-SIAM, 2004. 12. G. Owen. Values of graph-restricted games. SIAM Journal on Algebraic and Discrete Methods, 7(2):210-220, 1986. 13. N. Robertson and P.D. Seymour. Graph minors II. Algorithmic aspects of tree width. Journal of Algorithms, 7:309-322, 1986. 14. D.J. Rose. Triangulated graphs and elimination process. J. Math. Analys. AppL, 32:597-609, 1970. 15. D.J. Rose, R.E. Tarjan, and G.S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing, 5:266-283, 1976. 16. L.S. Shapley. A value for n-person games. In H.W. Kuhn and A.W. Tucker, editors. Contributions to the Theory of Games II, pages 307-317, Princeton, 1953. Princeton University Press. 17. R.E. Tarjan and M. Yannakakis. Simple linear-time algorithms to test chordality of graphs, acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing, 13:566-579, 1984. 18. R. van den Brink and P. Borm. Digraph competitions and cooperative games. Theory and Decision, 53:327-342, 2002. 19. M. Wiegers. Recognizing outerplanar graphs in linear time. In Proceedings of Graph-Theoretical Concepts in Computer Science, volume 246 of LNCS, pages 165-176. Springer-Verlag, 1987. 20. K. Zilles. Architecture of the Human Cerebral Cortex. Regional and Laminar Oganization. In G. Paxinos and J.K. Mai, editors, The Human Nervous System, pages 997-1055, San Diego, CA, 2004. Elsevier. 2nd edition.

On PTAS for Planar Graph Problems Xiuzhen Huang ^ a n d Jianer Chen'^ ^ Department of Computer Science, Arkansas State University, State University, Arkansas 72467. Email: [email protected] ^ Department of Computer Science, Texas A&M University, College Station, TX 77843. Email: [email protected]** A b s t r a c t . Approximation algorithms for a class of planar graph problems, including PLANAR INDEPENDENT SET, PLANAR VERTEX COVER and

PLANAR DOMINATING SET, were intensively studied. The current upper bound on the running time of the polynomial time approximation schemes (PTAS) for these planar graph problems is of 2°'-''/^'n°^^\ Here we study the lower bound on the running time of the PTAS for these planar graph problems. We prove that there is no PTAS of time 20(\/1A)„0(1) £QJ. PLANAR INDEPENDENT SET, PLANAR VERTEX COVER and PLANAR DOMINATING SET unless an unlikely collapse occurs in parameterized complexity theory. For the gap between our lower bound and the current known upper bound, we specifically show that to further improve the upper bound on the running time of the PTAS for PLANAR VERTEX COVER, we can concentrate on PLANAR VERTEX COVER on pla-

nar graphs of degree bounded by three.

1 Introduction There is intensive research work on a class of planar graph N P - h a r d optimization problems, such as PLANAR I N D E P E N D E N T S E T , P L A N A R V E R T E X C O V E R a n d

PLANAR DOMINATING SET. Approximation algorithms for these planar graph problems and related problems were studied by researchers such as Bar-Yehuda and Even [5], Lipton a n d Tarjan [25], Baker [4], Eppstein [16], Grohe [20], K h a n n a and Motiwani [24], and Cai et al. [7]. T h e current upper bound on t h e running time of t h e polynomial time approximation scheme (PTAS) for these planar graph problems is of 2°(^/^)n'-'^^' [4, 25]. In this paper, we study the lower bound on t h e running time of t h e P T A S algorithms for these planar graph problems. O u r work follows some recent research progress in parameterized complexity theory [10, 11], where strong computational lower bound results on the running time of the algorithms for W[^]-hard problems are derived, t > 1. This research is supported in part by US NSF under Grants CCR-0311590 and CCF-0430683. Please use the following format when citing this chapter: Huang, X., Chen, J., 2006, in International Federation for Information Processing, Volume 209, Fourth IFIP International Conference on Theoretical Computer Science-TCS 2006, eds. Navarro, G., Bertossi, L., Kohayakwa, Y., (Boston: Springer), pp. 299-313.

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Our research work here is focused on the computational lower bounds on the running time of the algorithms for the parameterized problems that are fixedparameter tractable (in FPT). We first give a brief review on parameterized complexity theory and the recent research results in [10, 11]. A parameterized problem Q is a decision problem consisting of instances of the form {x,k), where the integer fc > 0 is called the parameter. The parameterized problem Q is fixed-parameter tractable [15] if it can be solved in time f{k)\x\'^^^\ where / is a recursive function^. Certain NP-hard parameterized problems, such as VERTEX COVER, are fixedparameter tractable, and hence can be solved practically for small parameter values [12]. On the other hand, the inherent computational difficulty for solving many other NP-hard parameterized problems with even small parameter values has suggested that certain parameterized problems are not fixed-parameter tractable, which has motivated the theory oi fixed-parameter intractability [15]. The ly-hierarchy lJj>o W[t] has been introduced to characterize the inherent level of intractability for parameterized problems. A large number of parameterized problems have been proved to be hard or complete for various levels in the VF-hierarchy [15]. Examples of iy[l]-hard problems include many wellknown NP-hard problems such as CLIQUE, DOMINATING SET, SET COVER, and WEIGHTED CNF SATISFIABILITY. The theory of parameterized intractability has found important applications in a variety of areas such as database systems and model checking [20, 27]. The M^[l]-hardness of a parameterized problem provides a strong evidence that the problem is not fixed-parameter tractable, or equivalently, cannot be solved in time f{k)n'-"'^^ for any function / . Recent investigation has derived much stronger computational lower bounds on the running time of the algorithms for well-known NP-hard parameterized problems [10, 11]. For example, it has been shown that unless an unlikely collapse occurs in the parameterized complexity theory, any algorithm solving the iy[l]-hard CLIQUE problem takes time at least n^^''\ Note that this lower bound is asymptotically tight in the sense that the trivial algorithm that enumerates all subsets of k vertices in a given graph to test the existence of a chque of size k runs in time 0{n''). Similar lower bound results could be shown for other VF[i]-hard problems, t > 1. A method for deriving lower bounds on the running time of approximation algorithms for NP-hard combinatorial optimization problems is designed. It was proved in [11] that unless an unlikely collapse occurs in parameterized complexity theory, the VF[l]-hardness of the parameterized problem under the linear fpt-reduction implies the nonexistence of polynomial time approximation schemes of running time f{\/e)n°^^/'^^ for the original optimization problem, where / is any recursive function. ^ In this paper, we always assume that complexity functions are "nice" with both domain and range being non-negative integers and the values of the functions and their inverses can be easily computed. For two functions / and g, we write /(n) = o{g{n)) if there is a nondecreasing and unbounded function A such that /(n) < g{n)/\{n). A function / is subexponential if /(n) = 2°^"^

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2 Terminologies in Approximation For a reference of the theory of approximation, the readers are referred to the book [3]. In this section, we provide some basic terminologies for studying approximability and its relationship with parameterized complexity. An NP optimization problem Q is a four-tuple {IQ,SQ, fQ,optQ), where 1. JQ is the set of input instances. It is recognizable in polynomial time; 2. For each instance x G IQ, SQ{X) is the set of feasible solutions for x, which is defined by a polynomial p and a polynomial time computable predicate n {p and TT only depend on Q) as SQ{X) = {y : \y\ < p{\x\) and iT{x,y)}; 3. fQ{x,y) is the objective function mapping a pair x G IQ and y £ SQ{X) to a non-negative integer. The function fg is computable in polynomial time; 4. optq € {max, min}. Q is called a maximization problem if optg = max, and a minimization problem if optg = min. An optimal solution yo for an instance x £ / Q is a feasible solution in SQ{X) such that fQ{x,yo) = optQ{fQ{x,z) | z £ SQ{X)}. We will denote by optQ{x) the value optQ{fQ{x,z) \ z € SQ{X)}. An algorithm A is an approximation algorithm for an NP optimization problem Q = {IQ, SQ,fQ,optQ) if, for each input instance x in IQ, A returns a feasible solution yA{x) in SQ{X). The solution yA{x) has an approximation ratio r{n) if it satisfies the following condition: optQ{x)/fQ{x,yA{x)) fQ{x,yA{x))/optQ{x)

< r{\x\) if Q is a maximization problem < r{\x\) if Q is a minimization problem

The approximation algorithm A has an approximation ratio r{n) if for any instance x in IQ, the solution yA{x) constructed by the algorithm A has an approximation ratio bounded by r(|a:|). Definition 1. An NP optimization problem Q has a polynomial-time approximation scheme (PTAS) if there is an algorithm AQ that takes a pair {x,e) as input, where x is an instance of Q and e > 0 is a real number, and returns a feasible solution y for x such that the approximation ratio of the solution y is bounded by 1 -\- e, and for each fixed e > 0, the running time of the algorithm AQ is bounded by a polynomial of \x\.* An NP optimization problem Q has a fully polynomial-time approximation scheme (FPTAS) if it has a PTAS AQ such that the running time of AQ is bounded by a polynomial of \x\ and 1/e. "* There is an alternative definition for PTAS in which each e > 0 may correspond to a different approximation algorithm Ae for Q [19]. The definition we adopt here may be called the uniform PTAS, by which a single approximation algorithm takes care of all values of e. Note that most PTAS developed in the literature are uniform PTAS.

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Observe that the time complexity of a PTAS algorithm may be of the form 0(2^/^|a:|'^) for a fixed constant c or of the form 0(1x1^/"^). Obviously, the latter type of computations with small e values will turn out to be practically infeasible. This leads to the following definition [9]. Definition 2. An NP optimization problem Q has an efficient polynomial-time approximation scheme (EPTAS) if it admits a polynomial-time approximation scheme whose time complexity is bounded by 0(/(l/e)|a;|'^), where f is a recursive function and c is a constant. An NP optimization problem Q can be parameterized in a natural way as follows. Definition 3. Let Q = {lQ,SQ,fQ,optQ) be an NP optimization problem. The parameterized version of Q is defined as follows: (1) If Q is a maximization problem, then the parameterized version of Q is defined as Q> = {{x, k) | a; € / Q A optQ{x) > k]; (2) If Q is a minimization problem, then the parameterized version of Q is defined as Q< = {{x,k) | x € IQ A optQ{x) < k). The above definition offers the possibility to study the relationship between the approximability and the parameterized complexity of NP optimization problems. However, there is an essential difference between the two categories: an approximation algorithm for an NP optimization problem constructs a solution for a given instance of the problem, while a parameterized algorithm only provides a "yes/no" decision on an input. To make the comparison meaningful, we need to extend the definition of parameterized algorithms in a natural way so that when a parameterized algorithm returns a "yes" decision, it also provides an "evidence" to support the conclusion (see [6] for a similar treatment). Definition 4. Let Q = {IQ,SQ, fQ,optQ) be an NP optimization problem. We say that a parameterized algorithm AQ solves the parameterized version of Q if (1) in case Q is a maximization problem, then on an input pair {x, k) in Q>, the algorithm AQ returns "yes" with a solution y in SQ{X) such that fQi^^y) ^ k> o,nd on any input not in Q>, the algorithm AQ simply returns "no"; (2) in case Q is a minimization problem, then on an input pair {x, k) in Q<, the algorithm AQ returns "yes" with a solution y in SQ{X) such that fQ{x,y) < k, and on any input not in Q<, the algorithm AQ simply returns "no".

3 Lower Bound on Running Time of P T A S for P l a n a r G r a p h Problems Suppose e > 0 is the given error bound, and n is the number of vertices of a planar graph. Lipton and Tarjan [25] designed an EPTAS approximation

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algorithm of time 0(2°(^/'')n°(^)) for PLANAR INDEPENDENT SET, as an application of a separator theorem on planar graphs. Based on the outer-planarity of planar graphs, Baker [4] designed EPTAS algorithms of time 0{2'-'^^/'^^n) for several famous NP-hard optimization problems on planar graphs, such as PLANAR VERTEX COVER, PLANAR INDEPENDENT SET, and PLANAR DOMINATING SET. In [6], Cai and Chen proved that if an optimization problem has a fully polynomial-time approximation scheme (FPTAS), then the corresponding parameterized problem is fixed-parameter tractable (in FPT). Later this result was extended in [9] by Cesati and Trevisan: All optimization problems that have efficient polynomial time approximation schemes (EPTAS) have their parameterized problems in FPT. Therefore, the parameterized versions of these aforementioned optimization problems, PLANAR VERTEX COVER, PLANAR INDEPENDENT SET, and PLANAR DOMINATING SET, are in FPT. Alber et. al [2] designed parameterized algorithms of time 2'^^^''^n'^^^^ for the parameterized versions of the above NP-hard optimization problems. A lot of research has been done on these problems to try to further improve the time complexity of the parameterized algorithms. Interested readers are referred to [1, 23, 17, 18]. Cai et. al [8] proved the following lower bound result for the parameterized algorithms of these problems: Lemma 1. (Lemma 5.1 in [8]) PLANAR VERTEX COVER, PLANAR INDEPENDENT SET, and PLANAR DOMINATING SET do not have parameterized algorithms oftime2°^^^n'^^'^\ unless VERTEX COVER-3 has 2"'^''^n'-"^'^^-time parameterized algorithms. The class SNP introduced by Papadimitriou and Yannakakis [26] contains many well-known NP-hard problems including, for any fixed q > 3, CNF q-SAT, q-COLORABILITY, q-SET COVER, and VERTEX COVER, CLIQUE, and INDEPENDENT SET [22]. It is commonly believed that it is unlikely that all problems in SNP are solvable in subexponential time. Impagliazzo, Paturi and Zane [22] studied the class SNP and identified a group of SNP-complete problems under the serf-reduction, such that if any of these SNP-complete problems is solvable in subexponential time, then all problems in SNP are solvable in subexponential time. This group of SNP-complete problems under the serf-reduction includes t h e p r o b l e m s CNF q-SAT, q-COLORABILITY, q-SET COVER, a n d VERTEX COVER, CLIQUE, a n d INDEPENDENT SET.

We have: Lemma 2. (Theorem 3.3 in [13]) The VERTEX COVER-3 problem can he solved in 2°('^)n'^'^^^ time if and only if the VERTEX COVER problem can be solved in 2o(fc)^o(i) ^-^g^

Therefore Lemma 1 could be restate as:

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L e m m a 3 . PLANAR VERTEX COVER, PLANAR INDEPENDENT SET, anrf PLANAR

DOMINATING SET do not have parameterized algorithms of time 2°^^'°)n'-'^^^, unless all SNP problems are solvable in subexponential time. We prove the following lower bound results on the running time of the EPTAS algorithms for those planar graph problems: T h e o r e m 1. PLANAR VERTEX COVER, PLANAR INDEPENDENT SET, and PLA-

NAR DOMINATING SET have no EPTAS of running time 2°(VVe)n'-'(^), where e > 0 is the given error bound, unless all SNP problems are solvable in subexponential time. Proof We provide the proof for

PLANAR VERTEX COVER.

Let Q be the mini-

mization problem of PLANAR VERTEX COVER.

Prom the EPTAS algorithm AQ for the PLANAR VERTEX COVER problem Q, we provide the parameterized algorithm A< shown in Fig. 1 for the parameterized version Q< of the PLANAR VERTEX COVER problem Q. Algorithm A<: Input: An instance (G, k) of Q<, where G is a planar graph. Output: If the minimum vertex cover Go has the size |Go| < k, then Output "yes"; otherwise Output "no". 1. On the instance {G,k) of Q<, call the EPTAS algorithm AQ on G and e = l/(2fc + 1). Suppose that the algorithm AQ returns a vertex cover G. 2. If \C\ < k, then return "yes"; otherwise return "no".

Fig. 1. Algorithm A<. We verify that the algorithm A< solves the parameterized problem Q<. Since the PLANAR VERTEX COVER problem Q is a minimization problem, if \C\ < k then obviously |Co| < k. Thus, the algorithm A< returns a correct decision in this case. On the other hand, suppose \C\ > k. Since \C\ is an integer, we have \C\> k + 1. Since AQ is a EPTAS for the PLANAR VERTEX COVER problem Q and e = l/(2fc -f-1), we must have

|C|/|(7oifc+ 1) |Co| > |C|/(1 + l/(2fc + 1) > (fc + 1)/(1 + l/(2fc + 1) =fc+ 1/2 > fc Thus, in this case the algorithm A< also returns a correct decision. This proves that the algorithm A< solves the parameterized version Q< of the PLANAR

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VERTEX COVER problem Q. The running time of the algorithm A< is dominated by that of the algorithm AQ, which is bounded by 2°^Vy'^)n^W = 2°(^)n'^W. Thus, the parameterized version Q< of the PLANAR VERTEX COVER problem is solvable in time 2°^^''^n'-^^^\ Therefore, the result in the theorem follows from Lemma 3. T h e p r o o f s for PLANAR INDEPENDENT SET a n d PLANAR DOMINATING SET

are similar and hence are omitted. C o r o l l a r y 1. PLANAR VERTEX COVER, PLANAR INDEPENDENT SET, and

PLA-

NAR DOMINATING SET have no PTAS of running time 2 ° ( v ^ ) n ° ( ^ ) , where € > 0 is the given error bound, unless all SNP problems are solvable in subexponential time. By a comparison with the upper bound on the running time of the EPTAS algorithms for these planar graph problems in Baker [4], which is 2^^^^'^^nP^^^ (also in Lipton and Tarjan [25]), we can see that there is a gap between the upper bound result and our lower bound result in Theorem 1. To come up with new approaches to improve the upper bound on the running time of the EPTAS algorithms in [4] will be interesting research. To study this issue, we concentrate on the PLANAR VERTEX COVER problem in the next section.

4 U p p e r Bound on Running Time of P T A S for P l a n a r Vertex Cover In this section, we study the PTAS algorithms for the VERTEX COVER problem on planar graphs of degree bounded by 3, abbreviated as P-vc-3. The VERTEX COVER problem on general planar graphs is abbreviated as P-VC. Prom the proof of Theorem 1, we get the following lemma: L e m m a 4. The P-vc-3 problem has no EPTAS of running time 2''(viA)n°(^), where e > 0 is the given error bound, unless the P-vc-3 problem has a parameterized algorithm of time 2°'^^^^n'~"^^\ It is well known that a planar embedding of a planar graph can be constructed in linear time [21]. We define an operation, called the unfolding operation, based on a planar embedding of a planar graph. Definition 5. Suppose that G is a planar graph with a planar embedding i^{G), and that v is a degree-d vertex in G, where d > 3, with neighbors vi, v^, • • •, Vd, such that when one traverses around the vertex v on the embedding 7r(G), the edges incident to v are in the cyclic order [v,vi], [v,V2], •••, [v,Vci]. The unfolding operation on the vertex v will do the following: remove the vertex v from TT{G), and add a path of length 2d — 5: Pv = {yi,xi,y2,X2,

•••

,yd-3,Xd-3,yd-2}

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where each vertex Xi is of degree 2 and adjacent to the vertices yi and yi+i, and each vertex yi is of degree 3 such that j/i is adjacent to {ui, i;2, Xi}, yd-2 is adjacent to {vd-i,V(i,Xd.-3}, and yi is adjacent to {vi+i,Xi-i,Xi}, for 2 3, where y<3 is the set of vertices whose degree is less than or equal to 3, y>3 is the set of vertices whose degree is greater than 3. We apply the unfolding operation on a vertex v G V>3. We get a new planar graph G2 = (V2, E2), where G2 has one fewer vertex of degree larger than 3, compared with Gi. We first consider a vertex cover C2 of the graph G2. - Suppose for some i, 1 < i < d — i, the three vertices Xj, j/j, and yj+i are all in C2. Then we simply remove Xi from C2. It is obvious that C2 — {xi} is still a vertex cover of G2, with one fewer vertex compared with C2. Call this operation clean-one. - Suppose for some i, 1 < i < d — 3, exactly two of the three vertices Xi, yi, and j/j+i are in C2- If one of these two vertices is Xi, then we can replace the two vertices by j/j and j/j+i, resulting in a new vertex cover of the same size. Call this operation clean-two.

V3

V

Vl

V2

Fig. 2. Unfolding operation on the vertex v (with degree 6). Note that at least one of the three vertices Xi, y^, and yj+i must be in the vertex cover C2 in order to cover the edges [xi,yi] and [xi,yi+i]. Therefore, besides the above cases, the only remaining case is that for the three vertices Xi, yi, and y^+i, only one of them is in C2. In this case, this vertex in C2 must b e Xi.

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In the following discussion, cleaning a vertex cover C2 means that we apply the processing of clean-one and clean-two on C2. After the cleaning process, we say that the vertex cover C2 is clean. By the above discussion, in a clean vertex cover C2 of the graph G2, we have Claim. Either all d — 3 vertices Xi, 1 < i < d — 3, are in C2 and none of the d — 2 vertices yj, 1 < j < d — 2, is in C2; or all d — 2 vertices yj, 1 < j < d — 2, are in C2 and none of the d — 3 vertices Xi, 1 < i < d — 3, is in C2. Let Ci be any vertex cover of the graph Gi such that Ci has ki vertices. If V G Ci (so V covers the d edges [^,^1], ..., [v,Vd] in G), then by replacing v in Ci by the d—2 vertices j/i, 2/2, • • •, yd-2 in G2, we obviously get a clean vertex cover C2 for the graph G2. The vertex cover C2 has ki + {d — 3) vertices. On the other hand, if v is not in Ci (so the edges [f, vi], ..., [v, Vd] must be covered by the vertices vi, ..., v^ in Ci), then by adding the d — 3 vertices xi, X2, • • •, Xd-3 to Ci, we get a clean vertex cover C2 for the graph G2 and C2 contains k\ + {d — 3) vertices. In conclusion, from a vertex cover of fei vertices for the graph Gi, we can always construct a (clean) vertex cover of k\ + [d — 3) vertices for the graph G2. Conversely, suppose that we are given a clean vertex cover C2 of the graph G2, where C2 has k2 vertices. If C2 contains the d — 2 vertices j/i, 2/2, • • •, yd-2, then replacing the d — 2 vertices yi, y2, ..., yd-2 in C2 by a single vertex v gives a vertex cover of k2 — {d — 3) vertices for the graph Gi. On the other hand, if C2 contains the d — 3 vertices xi, X2, . •., Xd-3, then removing these d — 3 vertices from C2 gives a vertex cover of k2 — {d — 3) vertices for the graph Gi. In conclusion, from a vertex cover of ^2 vertices for the graph G2, we can always construct a vertex cover of k2 — [d- 3) vertices for the graph Gi. Now suppose that the set of vertices of degree larger than 3 in the graph Gi is y>3 = {ui,U2,.. • ,Ur-}. Denote by deg{u) the degree of the vertex u. Inductively, suppose that the graph Gj+i is obtained from the graph G, by unfolding the vertex Uj, for 1 < i < r. Note that the graph Gr has its degree bounded by 3, and we say that the graph Gr is obtained from the graph Gi by unfolding all vertices of degree larger than 3. Let C\ be a vertex cover for the graph Gi with \Gi\ = ki. By the above discussion, we can construct from Gi a vertex cover G2 of ki + {deg{ui) — 3) vertices for the graph G2; then from G2, we can construct a vertex cover G3 of ki + (deg{ui) — 3) -f {deg(u2) — 3) vertices for the graph G3, , and finally we construct a vertex cover Cr of fci -t- '}2\^i{deg{ui) — 3) vertices for the graph GrOn the other hand, let Cr be a vertex cover of kr vertices for the graph GrFirst we clean Cr to get a clean vertex cover C'r for Gr. Since cleaning does not increase the size of the vertex cover, we have |G^| < \Cr\ = kr. Now by the above discussion, we can get a vertex cover Cr-i of |C^| — {deg{ur) — 3)
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{deg{ur-i) — 3) vertices for the graph Gr-2, , finally, we will construct a vertex cover of at most kr — ^l^i{deg{ui) — 3) vertices for the graph G\. In particular, the above discussion enables us to derive a relation between the minimum vertex covers for the graphs G\ and Gr- Let h\ and k^ be the sizes of minimum vertex covers of the graph G\ and Gr, respectively. By the above discussion, from a minimum vertex cover for the graph G\, we can construct a vertex cover of k\ + ^[=i(c?e(7(w,) — 3) vertices for the graph Gr- Therefore, fci + X)[=i(^65(^i) ~ 3) > ky. On the other hand, from a minimum vertex cover of the graph G^, we can construct a vertex cover of no more than kr — Si=i('^65(^i) ^ 3) vertices for the graph G\, thus kr — J2l=i(.deg{ui) — 3) > fci. Combining these two relations, we get fci + YA=i{deg{ui) — 3) = krSummarizing the above discussion, we get the following: Claim. Let Gi be a graph in which the set of vertices of degree larger than 3 is V'>3. Let Gr be a graph obtained by unfolding all vertices of degree larger than 3 in Gi. Then from a vertex cover Ci for the graph Gi, we can construct in polynomial time a vertex cover of |Gi| + Yluev id^9i'^) " 3) vertices for the graph Gr', and from a vertex cover Cr for the graph Gr, we can construct in polynomial time a vertex cover of at most \Cr\ — X)uev ideg{u) — 3) vertices for the graph Gi. Moreover, the size of a minimum vertex cover of the graph Gr is equal to the size of a minimum vertex cover of the graph Gi plus Using the unfolding operations, we can prove Lemma 5. The P-vc-3 problem has no parameterized algorithm of time 2°^^^^n'-'^^\ unless the P-VC problem has a parameterized algorithm of time 2°^'^''^n'^^^\ Proof. Suppose the P-VC-3 problem has a parameterized algorithm A of time 20(^^)7^0(1). We have the following algorithm A' shown in Fig 3 for the P-VC problem. We prove the algorithm A' is correct. By Claim 4, OPTi is a vertex cover for the graph Gi with \0PT2\ — J2uev i^^di'^) " 3) vertices and OPTi is computable in time n'-'^^\ Since OPT2 is a minimum vertex cover for the graph G2, by Claim 4 again, a minimum vertex cover for the graph Gi contains IOPT2I — X^^jgy {deg{u) — 3) vertices. In conclusion, OPTi is a minimum vertex cover for the graph Gi. We analysis the running time of A' in the following. For the graph Gi = {Vi,Ei), Vi = V<3 Uy>3, where l^il = n and \Ei\ = m, we can always assume \OPTi\ > n/2 by applying the NT-theorem [12]. That is, the parameter k > n/2. After applying the unfolding operation on each V G V>3, we get the new planar graph G2 = (V2, i?2) with degree bounded by 3. The construction of G2 can be done in polynomial time. For a planar graph with n vertices and m edges, we have [14]: m < 3n — 6.

(1)

On PTAS for Planar Graph Problems

309

Algorithm A' Input: A planar graph Gi = {Vi,Ei), Vi = V<3UV>3, and an integer k > 0. Output: Output "Yes", if the size of the minimum vertex cover OPTi of Gi satisfies jOPTil 3 be the set of all vertices of degree larger than 3 in the graph Gi. Construct a planar graph G2 by unfolding all vertices of degree larger than 3 inGi. 2. Run the algorithm A on the graph G2 with the parameter ^2 = 1, 2,..., IV2I. We get a minimum vertex cover OPT2 for the graph G2. 3. Construct a vertex cover OPTi for the graph Gi from OPT2 such that \OPTt\ = IOPT2I - Zuev^Megiu) - 3). 5. If \OPTi\ < k, Return "Yes"; Otherwise, Return "No".

Fig. 3. Parameterized algorithm for PLANAR VERTEX COVER.

By Equation 1, for the graph Gi, the total degree of all its vertices satisfies: ^

deg(v) = 2m< 2(3n - 6) < 6n,

(2)

veVi

We have IV2I = \V<s\ + J2 ^(deg{v) - 3) + {deg{v) - 2))

V&V>3

<\Vi\+2Ydeg{v) veVi

n + 12n = 13n = 0(n). Therefore, the calls to the algorithm^ on the graph G2 takes time 2''^v'^^'^|V2|'^'^^ 2oiV^)j^oii) ^ 2°(^)n°(^). All the other steps of the algorithm A' takes polynomial time n'^^^\ Therefore the algorithm A' has running time 2°^^''^n^^^\ Therefore, from Lemma 4, Lemma 5 and Theorem 1, we have Theorem 2. The P-vc-3 problem has no EPTAS of running time 2"^^^"^ n'-"^^\ where e > 0 is the given error bound, unless all SNP problems are solvable in subexponential time.

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Theorem 2 implies t h e difficulty of improving t h e E P T A S algorithm for t h e P - v c - 3 problem. Baker [4] provided an E P T A S algorithm of time 2'^'^^/^'>p{n) for t h e P-VC problem. B y applying t h a t algorithm, we get an E P T A S algorithm of time 2'-'^^/^^p[n) for t h e P - v c - 3 problem. Since t h e P - v c - 3 problem seems simpler, one might suspect t h a t we could have a better E P T A S algorithm for it t h a n t h a t for t h e P-VC problem. In t h e following we show t h a t if we can improve t h e E P T A S algorithm for the P-VC-3 problem, then we can improve t h e E P T A S algorithm for t h e P-VC problem. T h e o r e m 3 . If the P-VC-3 problem has an EPTAS of running time then the P-VC problem has an EPTAS of running time f{13/e)n'-'^^\ is a recursive function and e > 0 is the given error bound.

f{l/e)n^^^\ where f

Proof Given an E P T A S algorithm A of running time f(l/e)n°^^^ for t h e P-VC3 problem, we provide an E P T A S algorithm B of running time /(13/e)n'^^^) for t h e P-VC problem. T h e description of algorithm B is given in Fig. 4. Algorithm B Input: A planar graph Gi = {Vi,Ei),

and a constant e > 0.

Output: A vertex cover Ci for Gi, such that |Gi| < (1 + e) * | O P r i | . 1. Let V>3 be the set of all vertices of degree larger than 3 in the graph G i . Unfold all vertices of degree larger than 3 in G i , let the resulting graph be G2 = {V2,E2), whose degree is bounded by 3. 2. Run the algorithm A with e' = e/13 on the graph G2. We get a vertex cover G2 for the graph G2. 3. From G2 construct a vertex cover Gi of at most IG2I — ^^^y vertices for the graph Gi.

('^sff('") " 3)

4. Return Gi.

Fig. 4. EPTAS algorithm for PLANAR VERTEX COVER.

We claim t h a t t h e vertex set C i is t h e required vertex cover for t h e graph By Equation 1 and Claim 4, we have

IOPT2I = lOPTil + Yl (^e5(w) - 3) u€V>3

u6Vi

On PTAS for Planar Graph Problems

311

< \OPTi\+&n < |(9PTi| + 12|OFTi| < 13|0PTi|. Therefore, \0PT2\ < 13|0PTi|.

(3)

By Claim 4, we have lOPTil = IOPT2I -

^

{deg{u) - 3)

USV>3

and •ueK>3

Therefore, we have |C2|-|Ci|>|OPT2|-|OPri! or equivalently |C2|-|OPT2|>|Ci|-|OPri| Prom this, we derive immediately |Ci|/|OPTi|-l = (|Ci|-|OPTi|)/|OPTi| <(|C2i-|OPT2|)/|OPTi| < 13(|C2| -

\OPT2\)/\OPT2\

= 13(|C2|/|OPr2| - 1) < 13*(e/13)

Here we have used the assumption that C2\/\OPT2\ < 1 + e' = 1 + e/13, and the fact IOPT2I > 13|0PTi|. The call of the algorithm A on the graph G2 takes time f{l/e')n^^^\ All the other steps of the algorithm B take polynomial time n'-'^^\ Therefore, the running time of the algorithm B is f{13/e)n'-^^^\ and the approximation ratio for the algorithm P is 1 + e.

5 Summary In this paper, we have proved lower bound results on the running time of the PTAS algorithms for a class of planar graph problems including PLANAR INDEPENDENT SET, PLANAR VERTEX COVER and PLANAR DOMINATING SET. We pointed out that there is a gap between our lower bound result and the current

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known upper bound result on t h e running time of the P T A S algorithms for these planar graph problems. We then studied t h e P T A S algorithms for PLANAR VERTEX COVER problem. Based on our study of t h e relationship between PLANAR VERTEX COVER and PLANAR VERTEX COVER on planar graphs of degree bounded by three, we showed t h a t to further improve the upper bound on the running time of the P T A S algorithms for P L A N A R V E R T E X C O V E R , we

could concentrate on t h e PLANAR VERTEX C O V E R on planar graphs of degree bounded by three. Closing t h e gap and further improving t h e upper bound on the running time of the P T A S algorithms for these planar graph problems are nice open problems inviting further research.

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16. Eppstein D (2000) Diameter and treewidth in minor-closed graph families, Algorithmica 27:275-291 17. Fomin FV and Thilikos DM (2003) Dominating sets in planar graphs: branchwidth and exponential speed-up. Proc. of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 168-177 18. Fomin FV and Thilikos DM (2004) A simple and fast approach for solving problems on planar graphs. Lecture Notes in Computer Science 2996:56-67 19. Garey M and Johnson D (1979) Computers and intractability: a guide to the theory of NP-Completeness. W. H. Freeman, New York 20. Grohe M (2003) Local tree-width, excluded minors, and approximation algorithms, Combinatorica 23:613-632 21. Hopcroft JE and Tarjan RE (1974) Efficient planarity testing. Journal of the ACM 21:549-568 22. Impagliazzo R, Paturi R, Zane F (2001) Which problems have strongly exponential complexity? Journal of Computer and System Sciences 63: 512-530 23. Kanj I, Perkovic L (2002) Improved parameterized algorithms for planar dominating set. Lecture Notes in Computer Science 2420:399-410 24. Khanna S, Motwani R (1996) Towards a Syntactic Characterization of PTAS, STOC 1996: 329-337 25. Lipton RJ, Tarjan RE (1980) Applications of a planar separator theorem. SIAM J. Comput. 9:615-627 26. Papadimitriou CH, Yannakakis M (1991) Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43: 425-440 27. Papadimitriou CH and Yannakakis M (1999) On the complexity of database queries. Journal of Computer and System Sciences 58:407-427

Index

Abraham, Marco 283 Arenas, Marcelo 3 Bockenhauer, Hans-Joachim Bloom, Stephen 231 Bortolussi, Luca 91 Brodnik, Andrej 103 Caromel, Denis 165 Chen, Jianer 299 Coja-Oghlan, Amin 271

Jeron, Thierry

251

Kotter, Rolf 283 Karlsson, Johan 103 Kiwi, Marcos 9 Kneis, Joachim 251 Kralovic, Rastislav 131 Krumnack, Antje 283 Kupke, Joachim 251 Kutrib, Martin 151 Lanka, Andre

d'Orso, Julien 213 Dean, Brian 65 Di Lena, Pietro 185 Dobrev, Stefan 131 Esik, Zoltan

231

Fabris, Prancesco 91 Flocchini, Paola 131 ForHzzi, Luca 251 Goemans, Michel 65 Goerdt, Andreas 271 Gruska, Jozef 5,17 Gutierrez, Claudio 7 Guttmann, Walter 77

197

271

Malcher, Andreas 151 Marchand, Herve 197 Matsuzaki, Kazutaka 115 Maucher, Markus 77 Munro, J. Ian 103 Nilsson, Andreas

103

Policriti, Alberto 91 Prencipe, Giuseppe 47 Proietti, Guido 251 Rusu, Vlad

197

Santoro, Nicola 11,47,131 Sei, Yuichi 115

Henrio, Ludovic 165 Honiden, Shinichi 115 Hromkovic, Juraj 251 Huang, Xiuzhen 299

Wanke, Egon 283 Widmayer, Peter 251

Immorlica, Nicole

Yannakakis, Mihalis

65

Touili, Tayssir

213

13