COMPOSITION OF STABLE SET POLYHEDRA

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1 COMPOSITION OF STABLE SET POLYHEDRA Benjamin McClosky and Illya V. Hicks Department of Comptational and Applied Mathematics Rice University November 30, 2007 Abstract Barahona and Mahjob fond a defining system of the stable set polytope for a graph with a ct-set of cardinality 2. We extend this reslt to ct-sets composed of a complete graph mins an edge and se the new theorem to derive a class of facets. 1 Preliminaries and Introdction Let G = (V, E) be a simple ndirected graph. A set S V of pairwise nonadjacent vertices defines a stable set. Let S := {S V S is a stable set}. Each S S has an incidence vector x S R V, where x S (v) = 1 if v S and x S (v) = 0 otherwise. Let P(G) be the convex hll of all x S sch that S S. P(G) is a fll-dimensional polytope and has a niqe (p to positive scalar mltiples) minimal defining system. Let V V and E(V ) := {v E, v V }. The sbgraph indced by V is G[V ] := (V, E(V )). For v V, define N G (v) := { V v E} and N G (v) := N G (v) {v}. A complete graph consists of pairwise adjacent vertices. A vertex set K V is complete whenever G[K] is a complete sbgraph. A maximal complete sbgraph defines a cliqe. A vertex v V is simplicial if G[N(v)] is complete. A ct-set C V decomposes G into a pair of proper sbgraphs (G 1, G 2 ) sch that C = V (G 1 ) V (G 2 ) and all paths from G 1 to G 2 intersect C. Chvátal [2] showed that the nion of defining systems of P(G 1 ) and P(G 2 ) defines P(G) when G[C] is a cliqe. Barahona and Mahjob [1] defined P(G) based on systems related to P(G 1 ) and P(G 2 ) when C 2. We extend this reslt to the case where G[C] is a complete graph mins an edge. Section 2 contains reslts necessary to extend Barahona and Mahjob s theorem. Section 3 generalizes their theorem. Section 3 refers to reslts from [1]. Section 4 applies the new theorem to derive a class of facets for the stable set polytope called diamonds. Section 5 smmarizes the reslts and discsses ftre research. bjm4@rice.ed, ivhicks@.rice.ed Department of Comptational and Applied Mathematics Rice University Hoston, 77005, USA. 1

2 v w 1 G k C\{,v} w 3 w 2 2 Spport Graphs Figre 1: The agmented graph G k. Sppose G has a ct-set C consisting of a nonadjacent pair of vertices. To obtain a defining system for P(G), Barahona and Mahjob [1] attach to C a new set of vertices {w i }. This agmentation defines a graph G. P( G) has a facet which projects along the sbspace of {w i } variables to define P(G). We generalize this method to the case where G[C] is a complete graph mins an edge. Section 3 analyzes the decomposition of G into the pair ( G 1, G 2 ). Here, we determine how the spport graphs of facets for P( G k ) interact with the {w i } vertices. Let a T x b be a nontrivial facet of P(G). Nontriviality implies b > 0 and a v 0 for all v V. In this section, all facets are assmed to be nontrivial. Define the following sets: V a := {v V a v > 0} and F a := {S S a T x S = b}. The spport graph of a T x b is defined as G a := G[V a ], the sbgraph indced by V a. Remark 1. Given a facet a T x b, F a consists of maximal stable sets in G a. In Section 3, we partition ineqalities based on their intersection with the set {w i }. Lemma 1 redces the nmber of partition sets. Recall that since P(G) is fll-dimensional, the sets S F a collectively satisfy no eqations other than scalar mltiples of a T x = b. Lemma 1. If a T x b is a non-cliqe facet, then G a contains no simplicial vertex. Proof. Sppose v V a is simplicial in G a. Then K := N Ga (v) is a cliqe and there exists an S F a sch that S K =. Otherwise, v K xs (v) = 1 for all S F a, a contradiction becase a T x b is not a cliqe ineqality. Observe that S is not a maximal stable set in G a, since S {v} is a feasible stable set. This contradicts Remark 1. Sppose G = (G 1, G 2 ) has a ct-set C where G[C] is a complete graph mins an edge. Notice G[C] has a stable set {, v}. For k {1, 2}, add the {w i } vertices to G k sch that N Gk (w 1 ) = {w 2 } (C \ {}), N Gk (w 2 ) = {w 1, }, and N Gk (w 3 ) = C. See Figre 1 for the agmented graph G k. The heavy edges denote joins (see [4]). For example, the edge between and C \ {, v} indicates that is adjacent to every vertex in C \ {, v}. Lemma 2. Let, v C be nonadjacent and C := C {w 1, w 2, w 3 }. For k {1, 2}, F k := {x P( G k ) z C x(z) = 2} is a facet for P( G k ). Moreover, no other facet contains all the vertices of C in its spport. 2

3 Proof. We show that F k is a facet for P( G k [ C]) by bilding a fll-rank C C matrix whose colmns are incidence vectors of all stable sets which lie on F k. See Figre 2. The first three rows correspond to w 1, w 2, and w 3. The last rows correspond to and C \ {}, respectively. Let 1 v be the ( C -1)-dimensional colmn vector with a 1 in row v and 0 s elsewhere. We now lift the ineqality z C x(z) 2 to a facet of P( G k ). Since all maximal stable sets J in G k satisfy J C = 2, the lifting coefficients for vertices in V ( G k ) \ C are zero. Ths, the ineqality is a facet of P( G k ). Sppose another facet a T x b contains all vertices of C in its spport. By Remark 1, z C xs (z) = 2 for all S F a. It follows that F k coincides with the face indced by a T x b. 1 0 T T T T I ( C 1) ( C 1) v Figre 2: The matrix A. Given a defining system for a polytope, the process of projecting along a sbspace of variables, say w 1 and w 2, is less complicated if the coefficients of w 1 and w 2 are binary. The following lemma allows the defining systems encontered in Section 3 to be pt in this form. Lemma 3 (Mahjob [3]). Given a facet a T x b, let w 1, w 2 V a be adjacent vertices in G a. If w 1 is simplicial in G a w 2 and w 2 is simplicial in G a w 1, then a w1 = a w2. Lemma 3 implies that a w1 = a w2 in any nontrivial facet containing both w 1 and w 2 in its spport. As a reslt, scaling these ineqalities by (1/a w1 ) = (1/a w2 ) will prodce ineqalities where both variables have binary coefficients. 3 Composition of Stable Set Polyhedra This section offers a straightforward extension of techniqes developed by Barahona and Mahjob. We will refer to reslts from [1]. Let G = (G 1, G 2 ) have a ct-set C where G[C] is a complete graph mins an edge. Constrct the agmented graph G by adding a new set of vertices {w i } to C, as in Section 2. Define C := C {w i }. P( G) has a facet F = {x P( G) z C x(z) = 2} sch that P(G) = proj w1,w 2,w 3 {F } = {x R G w R 3 s.t. (x, w) F }. The set C decomposes G into the pair ( G 1, G 2 ). In Section 2, it was shown that P( G k ) has a facet F k for k {1, 2}. 3

4 Lemma 4 (Barahona and Mahjob [1]). The facet F is defined by the nion of the systems that define F 1 and F 2. Lemma 4 relies on the existence of a fll-rank, sqare matrix of all incidence vectors for stable sets on F, F 1, and F 2. The matrix A constrcted in the proof of Lemma 2 (see Figre 2) implies that this lemma holds for the class of ct-sets C we are analyzing. In order to find a defining system for F, consider the defining system for P( G k ) (other than cliqe ineqalities involving the {w i } variables). Recall from Section 2 that the spport of a T x b is denoted by V a. Lemma 1 and Lemma 2 imply that the facet-defining ineqalities can be partitioned into three sets I k 1, I k 2,I k 3 defined as follows: I k 1 := {a T i x b i V ai {w 1, w 2, w 3 } = } I k 2 := {at i x b i V ai {w 1, w 2, w 3 } = {w 1, w 2 }} I k 3 := {at i x b i V ai {w 1, w 2, w 3 } = {w 3 }}. Let V k = V (G k ). Lemma 3 and Lemma 4 imply that the defining system of F can be written as follows, k {1, 2}: 1. j V k a k ij x(j) bk i, for all i Ik 1 2. j V k a k ij x(j) + x(w 1) + x(w 2 ) b k i, for all i Ik 2 3. j V k a k ij x(j) + x(w 3) b k i, for all i Ik 3 4. j C\ x(j) + x(w 1) 1 5. j C\ x(j) + x(w 3) 1 6. j C\v x(j) + x(w 3) 1 7. x() + x(w 2 ) 1 8. x(w 1 ) + x(w 2 ) 1 9. j C x(j) = x(j) 0, for all j Ṽk. The projection of this system along the sbspace of the {w i } variables is the polytope P(G). To define P(G), we proceed exactly as in [1]. Theorem 1. The polytope P(G) is defined by the nion of defining systems for P(G 1 ) and P(G 2 ), the non-negativity constraints, and the following facet-defining mixed ineqalities: a k ij x(j) + a l rj x(j) x(j) b k i + bl r 2 for k = 1, 2; l = 1, 2; k l; i Ik 2 ; r Il 3. j V k j V l j C Proof. See Theorem 3.5 and Corollary 3.7 in [1]. 4

5 4 Diamonds This section ses Theorem 1 to derive a class of facets for P(G). Let K 1,..., K 6 be sets of vertices sch that each K i is nonempty and complete. The graph G shown in Figre 3 is a member of a class of graphs which we call diamonds. The heavy edges denote joins. For example, an edge between K i and K j indicates that G[K i K j ] is complete. The size of the diamond is eqal to the nmber of sets K i. The diamond in Figre 3 has size 6, and z V x(z) 3 indces a facet for P(G). In general, facet-indcing diamonds have size 2n (where n > 1), a vertex sch that N G () = 2n i=1 K i, and a path P = p 1 p 2...p 2n 2 attached to the sets K 1,..., K 2n as shown in Figre 3. p 1 p 2 p 3 p 4 K 1 K 2 K 3 K 4 K 5 K 6 Figre 3: A diamond of size 6. Theorem 2. Let n > 1. If a diamond G has size 2n, then z V x(z) n indces a facet for P(G). Proof. The proof is by indction on n. Base case (n = 2): Choose v K 1 and w K 4. The diamond of size 4 has a 5-hole on the vertex set {p 1, p 2, w,, v}. Moreover, the odd-hole ineqality can be lifted to inclde all vertices in 4 i=1 K i. This implies that z V x(z) 2 indces a facet for P(G) as claimed. Indction step (n > 2): Sppose the theorem holds for all diamonds of even size less than 2n. The diamond of size 2n has a ct-set C = K 2n 3 {, p 2n 4 } which can be constrcted by removing an edge from a complete graph. Therefore, we apply Theorem 1. Figre 4 shows sbgraphs of the pair ( G 1, G 2 ). Let V 1 = V ( G 1 ) \ {w 1, w 2 } and V V ( G 2 ) \ {w 3 }. The graph on the left is a diamond of size 2n 2. By indction, 2 = x(z) n 1 (1) z V 1 is a facet for P( G 1 ). G2 has an odd-hole ineqality which lifts to obtain that x(z) 3 (2) z V 2 is a facet for P( G 2 ). Notice that ineqality (1) I3 1 and ineqality (2) I2 2. Theorem 1 gives the following facet-defining mixed ineqality for P(G) : x(z) + x(z) n z V (G 1 ) z V (G 2 )x(z) z C 5

6 p 1... p 2n-4 p 2n-4 p 2n-3 p 2n-2 K 1 K 2... K 2n-3 w 3 w 1 K 2n-3 K 2n-2 K 2n-1 K 2n w 2 Figre 4: Sbgraphs of G 1 and G 2. Upon simplifying, we obtain that z V x(z) n is a facet for P(G) as claimed. Theorem 2 fails when the diamond has size that is odd and at least three. To see this, let G be the diamond of size 3 shown in Figre 5. G is perfect and not a cliqe, so G is not a spport graph for any facet of P(G). Now let G be the diamond of size 5 also shown in Figre 5. If G is the spport graph of a facet, then there mst exist i=1 K i affinely independent maximal stable sets satisfying some eqation. However, no sch set exists. It follows by indction that a diamond of odd size is not a spport graph for any facet of P(G). 5 Conclsions and Ftre Work This paper generalized a theorem of Barahona and Mahjob concerning the composition of stable set polyhedra. The new reslt was applied to derive a class of facets called diamonds. Ftre research incldes sing composition to derive other classes of facets. An extension of Theorem 1 to more general ct-sets wold also be beneficial since composition can be applied recrsively. In other words, G can be decomposed into sbgraphs G 1,..., G m sch that the defining system for each P(G i ) is known. For example, decompose G into a set of perfect graphs. The defining systems for each P(G i ) can then be composed to define P(G). Another idea is to constrct P(G) starting from the leaves of a tree or branch decomposition. These approaches have the potential to characterize the stable set polytope for graphs which admit a strctred decomposition, bt they reqire a more general form of Theorem 1. Generalizing Theorem 1 might reqire techniqes different from the lift and project method of Barahona and Mahjob. Finding a G k and F k with the correct strctre ap- p p 1 p 2 p 3 K 1 K 2 K 3 K 4 K 5 Figre 5: Diamonds with odd size. 6

7 pears to be difficlt. A sbtle reqirement is that G k [ C] has to have exactly C affinely independent maximm stable sets. Otherwise, the matrix A is not invertible and Lemma 4 fails. Withot this restriction, Theorem 1 wold have held for any ct-set which partitions into two cliqes. There exist many graphs G k [ C] with exactly C affinely independent maximm stable sets, bt the ineqalities which define F k mst also involve the w i vertices in a strctred way. This strctre wold most likely involve extending the reslts of Section 2 to prevent the projection step from becoming too complicated. 6 Acknowledgements The athors wold like to thank the referee for comments which helped improve the presentation of this paper. This research was partially spported by NSF grants DMI and DMS References [1] F. Barahona and A.R. Mahjob, Compositions of Graphs and Polyhedra II: Stable Sets, SIAM J. Discrete Math., 7 (1994), pp [2] V. Chvátal, On Certain Polytopes Associated with Graphs, Jornal of Combinatorial Theory (B), 18 (1975), pp [3] A. R. Mahjob, On the Stable Set Polytope of a Series-Parallel Graph, Mathematical Programming, 40 (1988), pp [4] D. West, Introdction to Graph Theory, Prentice Hall, New York,