On a Representation of the Matching Polytope Via Semidefinite Liftings

Stephen, Tamon; Tunçel, Levent

doi:10.1287/moor.24.1.1

Cited by 44 publications

(30 citation statements)

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“…Stephen and Tunçel (1999) show that α(G) = n−1 2 iterations of the N + operator are needed for finding ST(G). Therefore, this gives an instance of a graph G for which ST(G) = Q α−1 (FR(G)) is strictly contained in N α−1 + (FR(G)).…”

Section: Application To the Stable Set Polytopementioning

confidence: 99%

A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0–1 Programming

2003

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Sherali and Adams (1990), Schrijver (1991) and, recently, Lasserre (2001) have constructed hierarchies of successive linear or semidefinite relaxations of a 0 − 1 polytope P ⊆ R n converging to P in n steps. Lasserre's approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments. We present the three methods in a common elementary framework and show that the Lasserre construction provides the tightest relaxations of P . As an application this gives a direct simple proof for the convergence of the Lasserre's hierarchy. We describe applications to the stable set polytope and to the cut polytope.

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Section: Application To the Stable Set Polytopementioning

confidence: 99%

A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0–1 Programming

2003

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“…Theorem 1.3 For n = 2d + 1, the Sherali-Adams rank of the matching polytope, in the worst case over all n-vertex graphs, is 2d − 1. Theorem 1.3 answers for the Sherali-Adams hierarchy a question initially posed by Lovász and Schrijver about the rank of the matching polytope in the LS + hierarchy, which was answered by Stephen and Tunçel [27].…”

Section: Introductionmentioning

confidence: 96%

“…For n = 2d + 1, they showed that the rank lies between 2d and 2d 2 − 1 in the LS hierarchy, and is at most d in the LS + hierarchy. Stephen and Tunçel [27] subsequently proved that the LS + -rank is exactly d, and Goemans and Tunçel [15] improved the upper bound on LS-rank to d 2 . Aguilera, Bianchi and Nasini [1] show that the LS-rank is strictly larger than d, and also that the rank in the weaker Balas-Ceria-Cornuéjols hierarchy is exactly d 2 .…”

Section: Introductionmentioning

confidence: 99%

Sherali-adams relaxations of the matching polytope

Mathieu

Sinclair

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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We study the Sherali-Adams lift-and-project hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1 + 1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rounds applied to the complete graph K 2d+1 . We give an explicit recurrence for the value of this LP, and hence show that its gap exhibits a "phase transition," dropping from close to its maximum value 1 + 1 2d to close to 1 around the threshold k = 2d − √ d. We also show that the rank of the matching polytope (i.e., the number of Sherali-Adams rounds until the integer polytope is reached) is exactly 2d − 1.

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“…For example, Stephen and Tunçel [29] showed that n iterations are needed for finding the matching polytope of K 2n+1 (starting with the polytope defined by nonnegativity and the degree constraints) using the N + operator. Recently, Cook and Dash [8] and Goemans and Tunçel [12] constructed examples where positive semidefiniteness does not help; namely, the same number d of iterations is needed for finding some d-dimensional polytope P using the N or the N + operator.…”

Section: Introduction Lovász and Schrijvermentioning

confidence: 99%

Tighter Linear and Semidefinite Relaxations for Max-Cut Based on the Lovász--Schrijver Lift-and-Project Procedure

Laurent¹

2002

SIAM J. Optim.

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Abstract. We study how the lift-and-project method introduced by Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166-190] applies to the cut polytope. We show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no K 5 -minor. Therefore, for a graph G with n ≥ 4 nodes with stability number α(G), n − 4 iterations suffice instead of the m (number of edges) iterations required in general and, under some assumption, n − α(G) − 3 iterations suffice. The exact number of needed iterations is determined for small n ≤ 7 by a detailed analysis of the new relaxations. If positive semidefiniteness is added to the construction, then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by Anjos and Wolkowicz [Discrete Appl. Math., to appear]. We also show how the Lovász-Schrijver relaxations for the stable set polytope of G can be strengthened using the corresponding relaxations for the cut polytope of the graph G ∇ obtained from G by adding a node adjacent to all nodes of G.Key words. linear relaxation, semidefinite relaxation, lift-and-project, cut polytope, stable set polytope AMS subject classifications. 05C50, 15A57, 52B12, 90C22, 90C27 PII. S1052623400379371 1. Introduction. Lovász and Schrijver [22] have introduced a method for constructing a higher dimensional convex set whose projection N (K) approximates the convex hull P of the 0-1 valued points in a polytope K defined by a given system of linear inequalities. If the linear system is in d variables, the convex set consists of symmetric matrices of order d + 1 satisfying certain linear conditions. A fundamental property of the projection N (K) is that one can optimize over it in polynomial time and thus find an approximate solution to the original problem in polynomial time. Moreover, after d iterations of the operator N , one finds the polytope P . Lovász and Schrijver [22] also introduce some strengthenings of the basic construction; in particular, adding positive semidefinite constraints leads to the operator N + , and adding stronger linear conditions in the definition of the higher dimensional set of matrices leads to the operators N and N + . They study in detail how the method applies to the stable set polytope. Starting with K = FRAC(G) (the fractional stable set polytope defined by nonnegativity and the edge constraints), they show that in one iteration of the N operator one obtains all odd hole inequalities (and no more), while in one iteration of the N + operator one obtains many inequalities including odd wheel, clique, and odd antihole inequalities and orthogonality constraints; therefore, the relaxation N + (FRAC(G)) is tighter than the basic semidefinite relaxation of the stable set polytope by the theta body TH(G). In particular, this method permits one to solve the maximum stable set problem in a t-perfect graph or in a perfect graph in polynomial time. They also show th...

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On a Representation of the Matching Polytope Via Semidefinite Liftings

Cited by 44 publications

References 3 publications

A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0–1 Programming

A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0–1 Programming

Sherali-adams relaxations of the matching polytope

Tighter Linear and Semidefinite Relaxations for Max-Cut Based on the Lovász--Schrijver Lift-and-Project Procedure

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