LOVÁSZ-SCHRIJVER SDP-OPERATOR AND A SUPERCLASS OF NEAR-PERFECT GRAPHS

Transcription

1 LOVÁSZ-SCHRIJVER SDP-OPERATOR AND A SUPERCLASS OF NEAR-PERFECT GRAPHS S. BIANCHI, M. ESCALANTE, G. NASINI, L. TUNÇEL Abstract. We study the Lovász-Schrijver SDP-operator applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the SDP-operator generates the stable set polytope in one step has been open since In an earlier publication, we named these graphs N + -perfect. In the current contribution, we propose a conjecture on combinatorial characterization of N + -perfect graphs and make progress towards such a full combinatorial characterization by establishing a new, close relationship among N + -perfect graphs, near-bipartite graphs and a newly introduced concept of full-support-perfect graphs. 1. Introduction In the early 1990 s Lovász and Schrijver [5] introduced the SDP relaxation N + (G) of STAB(G). Following the same line of reasoning used for perfect graphs, they proved that Maximum Weight Stable Set Problem can be solved in polynomial time for the class of graphs for which N + (G) = STAB(G). These graphs are called N + -perfect graphs [2]. To the best of our knowledge, no combinatorial characterization of Date: November 12, Key words and phrases. stable set problem, lift-and-project methods, semidefinite programming, integer programming. Partially supported by grants PID-CONICET 241, PICT-ANPCyT 0361, ONR N , and a Discovery Grant from NSERC. S. Bianchi: Universidad Nacional de Rosario, Argentina ( sbianchi@fceia.unr.edu.ar) M. Escalante: Universidad Nacional de Rosario, Argentina ( mariana@fceia.unr.edu.ar) G. Nasini: Corresponding author. Universidad Nacional de Rosario, Argentina ( nasini@fceia.unr.ac.ar) L. Tunçel: Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada ( ltuncel@math.uwaterloo.ca). 1

2 2 S. BIANCHI, M. ESCALANTE, G. NASINI, L. TUNÇEL N + -perfect graphs have been obtained so far. However, the set of N + - perfect graphs is known to contain many rich and interesting classes of graphs (e.g., perfect graphs, t-perfect graphs, wheels, anti-holes, nearbipartite graphs) and their clique sums. In the current contribution, we make some more progress towards a combinatorial characterization of N + -perfect graphs, by establishing a new, close relationship among N + - perfect graphs, near-bipartite graphs and a newly introduced concept of full-support-perfect graphs. A graph G is near-perfect if its stable set polytope is described by the clique inequalities and the full-rank constraint [7]. Clearly, perfect and minimally imperfect graphs are near-perfect. We say that a graph is properly near-perfect if it is an imperfect near-perfect graph. Using the results in [3] and [4] we deduce that all imperfect graphs with at most 6 nodes are also N + -perfect graphs, except for the two properly near-perfect graphs depicted in Figure 1, denoted by G LT and G EMN, respectively. These graphs prominently figure into our current work as the building blocks of an interesting family of graphs. Figure 1. The graphs G LT and G EMN. In this contribution, we focus on the problem of characterizing N + - perfect graphs. Based on all the work up to now, we propose the following conjecture: Conjecture 1. The stable set polytope of every N + -perfect graph G can be described by facet inducing inequalities of stable set polytopes of near-bipartite subgraphs of G. For a definition of near-bipartite graphs, see [8]. Establishing the correctness of the above conjecture is equivalent to showing that if a graph is N + -perfect and its stable set polytope has one full-support facet, then the graph is near-bipartite. In [2] we proved this result when the graph is near-perfect. In fact, we proved that if G is a properly

3 SDP-OPERATOR AND A SUPERCLASS OF NEAR-PERFECT GRAPHS 3 near-perfect N + -perfect graph then either G or its complement is an odd cycle. The main goal of the current contribution is to go further on this conjecture by studying N + -perfect graphs in a superclass of graphs including near-perfect graphs. This graph family was originally defined in [6] as the family of fs-perfect (full-support-perfect) graphs, together with some of their combinatorial properties. Actually, a graph is fsperfect if its stable set polytope is completely described by clique constraints and a single full-support facet. In this work, we prove that the graph operation called clique subdivision of an edge [1] preserves N + -imperfection. This result helps us prove that Conjecture 1 holds for fs-perfect graphs (Theorem 12). One of the main difficulties in obtaining a good combinatorial characterization for N + -perfect graphs is that the lift-and-project operator N + (and many related operators) can behave sporadically under wellstudied graph-minor operations (see [3, 4]). Therefore, in the study of N + -perfect graphs we are faced with the problem of constructing suitable graph operations and then deriving certain monotonicity or loose invariance properties under such graph operations. To arrive at our result, we start with analyzing a family of graphs obtained from taking the complete join of a node with a minimally imperfect graph. This structure on the one hand, generalizes wheels, and on the other hand, seems to capture an interesting class of minimally N + -imperfect graphs. In fact, we prove that every graph in this family with at least eight nodes is either a wheel or it can be obtained by a sequence of three graph operations (odd subdivision of an edge, 1-stretching of a node, clique subdivision of an edge) from the graph G LT or G EMN. Since each of these three graph operations preserves N + -imperfection (N + - rank is monotone nondecreasing under these operations), we conclude that the so-called properly fs-perfect graphs, with at least eight nodes, in the above family are N + -imperfect, with the exception of wheels. Finally, we show that every N + -perfect graph that is also properly fs-perfect must be the complete join of a minimally imperfect graph with a clique. This result leads to the fact that Conjecture 1 holds for fs-perfect graphs. 2. On minimal properly fs-perfect graphs We say that a graph is properly fs-perfect if it is fs-perfect but not near-perfect. Odd wheels are the simplest graphs which are properly fs-perfect. Wheels are particular cases of the complete join of graphs. Given two graphs G 1 and G 2, the complete join of them, denoted by

4 4 S. BIANCHI, M. ESCALANTE, G. NASINI, L. TUNÇEL G 1 G 2, is the graph having node set V (G 1 G 2 ) := V (G 1 ) V (G 2 ) and edge set E(G 1 G 2 ) := E(G 1 ) E(G 2 ) {vw : v V (G 1 ) and w V (G 2 )}. Given k 2, the 2k + 1-wheel, denoted by W, is the complete join of the trivial graph with only one node and the odd cycle C. Every facet defining inequality of STAB(G 1 G 2 ) can be obtained by the cartesian product of facets of STAB(G 1 ) and STAB(G 2 ). More precisely, ax β defines a facet of STAB(G 1 G 2 ) if and only if β = β 1 β 2 and a = (β 2 a 1, β 1 a 2 ) where a 1 x β 1 and a 2 x β 2 are facet defining inequalities of STAB(G 1 ) and STAB(G 2 ), respectively. Based on this property, we have the following lemma. Lemma 2. The complete join of two graphs is fs-perfect if and only if one of them is a complete graph and the other one is a properly fs-perfect graph. Proof. Assume that G = G 1 G 2 is a properly fs-perfect graph. Then, either G 1 or G 2 is imperfect. Without loss of generality, we may assume that G 2 is imperfect. From the properties of facets of STAB(G 1 G 2 ) discussed before the lemma, we conclude that G 1 has to be a complete graph. Since G 2 is an imperfect subgraph of G, it is either near-perfect or properly fs-perfect. Again using our remark on the facetal description of STAB(G 1 G 2 ), we conclude that G 2 is not nearperfect. Thus, G 1 is properly fs-perfect. Moreover, it is not hard to prove that a complete join of two graphs is N + -perfect if and only if both graphs are N + -perfect. This property together with Lemma 2 justify to focus our study on properly fs-perfect graphs which are not a complete join of two graphs. Let us first recall two well-known definitions: given a graph G with node set V and U V, G U, the graph obtained by deletion of U, is the subgraph of G induced by the nodes in V \ U. When U = {v} we simply write G v. If U is the set formed by a node v and all neighbours in G, we say that G U is the graph obtained by destruction of v and denote it by G v. Since every node induced subgraph of an fs-perfect graph is also fs-perfect, it is natural to consider minimal properly fs-perfect graphs, that is, properly fs-perfect graphs such that after deleting any node they loose the property of being properly fs-perfect. Clearly, every minimal properly fs-perfect graph has a minimally imperfect graph as a proper subgraph. This leads us to define F k as the family of graphs having node set {0, 1,..., 2k + 1} such that G 0 is a minimally imperfect

5 SDP-OPERATOR AND A SUPERCLASS OF NEAR-PERFECT GRAPHS 5 graph, for every k 2. The following result gives necessary conditions for a graph in F k to be fs-perfect. Theorem 3. [6] If G F k and G is a properly fs-perfect graph, then the following conditions hold: (1) α(g) = α(g 0); (2) α(g 0) α(g) 2; (3) the full support inequality Proof. Let (1) (α(g) α(g 0)) x 0 + defines a facet of STAB(G). i=0 a i x i β x i α(g) be the full support facet defining inequality of STAB(G) which can be assumed to have all integer coefficients a i 1 for every i = 0,...,. Since G 0 is a minimally imperfect graph, STAB(G 0) = { x QSTAB(G 0) : a i x i β Thus, the inequality a i x i β is a multiple of its rank constraint, i.e., there exists a positive integer number p such that a i = p for i = 1,..., 2k + 1 and β = p α(g 0). Therefore, (1) has the form (2) a 0 x 0 + p x i p α(g 0). Observe that there is at least one integer vector x satisfying (2) with equality and such that x 0 = 1. Then, a 0 + p x i = p α(g 0). Moreover, x is the incidence vector of a stable set S of G such that S {0} is a maximum stable set of G 0. Then, x i = α(g 0) }.

6 6 S. BIANCHI, M. ESCALANTE, G. NASINI, L. TUNÇEL and we obtain a 0 + p α(g 0) = p α(g 0) or equivalently, a 0 = p (α(g 0) α(g 0)). Therefore, the inequality (2) becomes (3) (α(g 0) α(g 0))x 0 + x i α(g 0). Since G is properly fs-perfect, we have that α(g 0) α(g 0) 2 or equivalently α(g 0) α(g 0) 2. To complete the proof, we only need to show that α(g) = α(g 0). Let x be the incidence vector of a maximum stable set in G, then (4) α(g) = x 0 + Moreover, by (3) x satisfies Since (α(g 0) α(g 0)) x 0 + x 0 + x i. x i α(g 0). x i (α(g 0) α(g 0)) x 0 + using (4), we have that α(g) α(g 0), implying α(g) = α(g 0). As a first consequence of the previous theorem we obtain: Corollary 4. Let G F k such that α(g) = 2. Then, G is fs-perfect if and only if G is near-perfect or G = {0} (G 0). Proof. By the previous theorem, α(g 0) α(g) 2 = 0 and hence, G = {0} (G 0). The converse follows from the definition of fs-perfect graphs. By Corollary 4, it suffices to consider fs-perfect graphs G in F k with α(g) 3. In this case, k 3 and G 0 = C. To complete our study of fs-perfect graphs, we make use of three operations in graphs. The first one is the well-known odd subdivision of an edge [9]. The second one is a type of stretching (1-stretching) defined in [2]. The third one is the clique subdivision of an edge, defined in [1]. These three operations allow us to conclude that every fs-perfect graph in the family F k with stability number at least three that is not a wheel, can be built from the graph G LT or graph G EMN. x i,

7 SDP-OPERATOR AND A SUPERCLASS OF NEAR-PERFECT GRAPHS 7 G 5 G' Figure 2. The graph G is obtained from G after the clique subdivision of edge [1, 2]. Let us now consider the graph operation defined in [1] which preserves N + -imperfection. Definition 5. [1] Given the graph G with nodes {1,..., n} and the clique (not necessarily maximal) K = {v 1,..., v s }, 2 s n in G, the clique subdivision of the edge [v 1, v 2 ] in K is defined as follows: G is obtained from G by deleting the edge [v 1, v 2 ], adding the nodes v n+1 and v n+2 together with the edges [v 1, v n+1 ], [v n+1, v n+2 ], [v n+2, v 2 ] and [v n+i, v j ] for i = 1, 2 and j = 3,..., s. Figure 2 illustrates the clique subdivision of the edge [1, 2] in the clique K = {1, 2, 3}. Notice that if the clique is K = {v 1, v 2 }, this operation reduces to the odd subdivision of [v 1, v 2 ]. Let us now present a result on the structure of the graphs in F k. Lemma 6. Let k 3 and G F k satisfy α(g) 3. Then, (1) G 0 is an odd hole; (2) if δ G (0) = 2k then G can be obtained after the clique subdivision of an edge in a graph G in F k 1 with δ G (0) = 2(k 1); (3) if δ G (0) 2k 1 and there are no consecutive nodes in G 0 with degree two then G can be obtained by 1-stretching of a node in a graph G F k 1 with δ G (0) 2k 2. Proof. (1) If G F k and α(g) 3 then as we observed above, G 0 = C and k = α(g) 3. (2) Let G be a graph in F k with δ G (0) = 2k. It is clear that there are four consecutive nodes in G 0 with degree 3. Then, G is a clique subdivision of a graph G in F k 1 with δ G (0) = 2(k 1). (3) If δ G (0) 2k 1 and there are two adjacent nodes of degree two, we may assume that these two nodes are 2k 1 and 2k +1. Let G be the graph in F k 1 having V (G ) = {0,..., 2k 1} and Γ G (0) = Γ G (0) {2k}. Then, Γ G (1) = {0, 2, 2k 1} and if we perform the 1-stretching operation on node 1 in G and

8 8 S. BIANCHI, M. ESCALANTE, G. NASINI, L. TUNÇEL name the two new nodes 2k and 2k + 1, we arrive at the graph G. Moreover, by construction, δ G (0) = δ G (0) 1. Theorem 7. Let k 3 and G F k be an fs-perfect graph different from W with α(g) 3. Then G can be obtained from G LT or G EMN after applying the odd subdivision of an edge, 1-stretching of a node or clique subdivision of an edge operations. Proof. The proof is by induction on k. Let us first analyze the case of k = 3. Since G is not the wheel, δ G (0) 6. (1) If δ G (0) = 6, the previous lemma implies that G is obtained after the clique subdivision of the graph G EMN. (2) If δ G (0) = 5 and the two nodes of degree two are adjacent, then G is an odd subdivision of the wheel graph W 5. If the nodes of degree two are nonadjacent then, using the previous lemma, G is obtained after the 1-stretching of G EMN. (3) If δ G (0) = 4 then, again, if two nodes of degree two are adjacent then G is an odd subdivision of G EMN. If not, it is a 1-stretching of G LT. (4) If δ G (0) = 3, there are always two adjacent nodes of degree two in G 0 and then G is an odd subdivision of G LT. For k 4, the result follows from the previous lemma by utilizing inductive arguments. 3. N + -perfect fs-perfect graphs It is known that the odd subdivision of an edge and the 1-stretching of a node operations preserve N + -imperfection (see [4] and [2], respectively). Based on the proof of Theorem 6.5 in [1], we have that the clique subdivision of an edge shares this property. First we give some details. In the proof of the result in [1], the authors define the sets (5) ˆK = K \ {v 1, v 2 } = {v 3,..., v s }, K = ˆK {vn+1, v n+2 }, K i = ˆK {v i, v n+i } for i = 1, 2. Using this definition they first show that: Lemma 8 ([1]). Let G be obtained from G by the clique subdivision of edge [v 1, v 2 ] in the clique K. Then, for x R n, let x R n+2 such that x i = x i for every i = 1,..., n, x n+1 = x 2 and x n+2 = x 1. (1) If x / STAB(G) then x / STAB(G ). (2) If x N k (G) then x N k (G ). Moreover, x( K) = x(k 1 ) = x(k 2 ) = x(k).

9 SDP-OPERATOR AND A SUPERCLASS OF NEAR-PERFECT GRAPHS 9 Now, we prove that the N + -rank does not decrease by clique subdivision of an edge. Theorem 9. Under the hypothesis of the previous lemma, we have that if x N k +(G) then x = (x, x 2, x 1 ) N k +(G ). Proof. Consider x N k +(G). The proof is by induction on k. Let k = 0. Then, the edge inequalities x n+1 + x n+2 = x 1 + x 2 1, x n+i + x j = x l + x j 1 for i = 1, 2, l = 1, 2, l j and j ˆK, are clearly satisfied. Now, let k 1 and let 1 x 1 x 2... x i... x 1 x x 1,i... x 2 0 x 2... x 2,i... Y =. x i x i,1 x i,2... x i.... be a representation of x N+(G). k Define 1 x 1 x 2... x i... x n+1 = x 2 x n+2 = x 1 x 1 x x 1,i... 0 x 1 x 2 0 x 2... x 2,i... x 2 0 Y. =. x i x i,1 x i,2... x i... x i,2 x i,1. x 2 0 x 2... x 2,i... x 2 0 x 1 x x 1,i... 0 x 1 By the induction hypothesis, column i belongs to x i cone(n+ k 1 (G )) for every i and the same holds for the difference between the first and the ith column, for every i. The symmetry of matrix Y follows from the symmetry of Y. In order to show that Y is PSD we prove that every submatrix Y S of Y with row and column set S {1,..., n + 2}, satisfies det(y S ) 0. If S {n + 1, n + 2} = then det(y S ) = det(y S) 0 since Y is PSD. If {2, n + 3} S or {3, n + 2} S then det(y S ) = 0. If S = S {n + 2} for S {1, 4, 5,..., n + 1} then det(y S ) = det(y T ) where T = S {3}. Similarly, if S = S {n + 3} for S {1, 4, 5,..., n + 1} then det(y S ) = det(y T ) where T = S {2}. Therefore, we have proved that x N+(G k ).

10 10 S. BIANCHI, M. ESCALANTE, G. NASINI, L. TUNÇEL Using the above lemma, we have Theorem 10. If G is obtained by clique subdivision of an edge in an N + -imperfect graph G then G is N + -imperfect. Proof. If N + (G) STAB(G), then there exists x N + (G) \ STAB(G). By Theorem 9, x = (x, x 2, x 1 ) N + (G ) and x / STAB(G ). The behaviour of the above-mentioned operations on N + -imperfect graphs and Theorem 7 allow us to prove the next result. Theorem 11. Let G be a properly fs-perfect graph in F k with k 3 and α(g) 3. Then, G is N + -perfect if and only if G = W. Proof. We only need to prove the only if part or equivalently that if G is not a wheel, then G is not N + -perfect. Since G is properly fs-perfect, G is not an odd subdivision of W 2s+1 for any s 3. By Theorem 7, if G is not a wheel, then it can be obtained from G LT or G EMN after a finite number of odd subdivisions of an edge, 1-stretching of a node and clique subdivisions of an edge operations. However, G LT and G EMN are N + -imperfect graphs and all the operations involved preserve N + -imperfection (we used Theorem 10). Hence, G is not N + - perfect. Finally, by using inductive arguments and Theorem 11, we are able to present the main result of this contribution. Theorem 12. Let G be a properly fs-perfect graph which is also N + - perfect. Then, G is the complete join of a complete graph and a minimally imperfect graph. Proof. Since G is a properly fs-perfect graph it has a minimally imperfect, node induced proper subgraph G. Let v V (G) \ V (G ) and G v the subgraph of G induced by {v} V (G ). Clearly, G v F r for some r 2 and G v is fs-perfect and N + -perfect. By Theorem 11, G v must be a wheel graph. Thus, we proved that, for every node v outside G, G v is the complete join of v and G. Therefore, G is the complete join of G, the subgraph of G induced by V (G) V (G ) and G. By Theorem 11, G is a complete graph and the theorem follows. Now, we can deduce that properly fs-perfect graphs which are also N + -perfect are near-bipartite. Moreover, they satisfy NB(G) = STAB(G). Therefore, based on the results obtained so far, we can conclude that Conjecture 1 holds for fs-perfect graphs.

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