Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut

Schoenebeck, Grant; Trevisan, Luca; Tulsiani, Madhur

doi:10.1145/1250790.1250836

Cited by 51 publications

(56 citation statements)

References 9 publications

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“…Schoenebeck, Tulsiani, and Trevisan [STT07b] show an integrality gap of 2 − ε remains after Ω(n) rounds of LS, which is optimal. This build on the previous work of Arora, Bollobas, Lovasz, and Tourlakis [ABL02,ABLT06,Tou06] who prove that even after Ω(log n) rounds the integrality gap of LS is at least 2 − ε, and that even after Ω((log n) 2 ) rounds the integrality gap of LS is at least 1.5 − ε.…”

Section: Previous Lower-bounds Workmentioning

confidence: 99%

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Linear Level Lasserre Lower Bounds for Certain k-CSPs

Schoenebeck

2008

2008 49th Annual IEEE Symposium on Foundations of Computer Science

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We show that for k ≥ 3 even the Ω(n) level of the Lasserre hierarchy cannot disprove a random k-CSP instance over any predicate type implied by k-XOR constraints, for example k-SAT or k-XOR. (One constant is said to imply another if the latter is true whenever the former is. For example k-XOR constraints imply k-CNF constraints.) As a result the Ω(n) level Lasserre relaxation fails to approximate such CSPs better than the trivial, random algorithm. As corollaries, we obtain Ω(n) level integrality gaps for the Lasserre hierarchy of 7 6 − ε for VertexCover, 2 − ε for k-UniformHypergraphVertexCover, and any constant for k-UniformHypergraphIndependentSet. This is the first construction of a Lasserre integrality gap.Our construction is notable for its simplicity. It simplifies, strengthens, and helps to explain several previous results.

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Section: Previous Lower-bounds Workmentioning

confidence: 99%

“…A more complete comparison can be found in [Lau03]. While there have been a growing number of integrality gap lower bounds for the LS [ABL02,ABLT06,Tou06,STT07b], the LS+[BOGH + 03, AAT05,STT07a,GMPT06], and the SA [dlVKM07,CMM07] hierarchies, similar bounds for the Lasserre hierarchy have remained elusive.…”

Section: Introductionmentioning

confidence: 99%

Linear Level Lasserre Lower Bounds for Certain k-CSPs

Schoenebeck

2008

2008 49th Annual IEEE Symposium on Foundations of Computer Science

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215

217

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“…However, we will not use this fact in the paper. The metric ρ π µ with π uv = (−1) d(u,v) was previously used in the integrality gap constructions by de la Vega and Kenyon-Mathieu [6] and Schoenebeck, Trevisan, and Tulsiani [18]. We denote this metric by ρ alt µ .…”

Section: Relaxations For Max Cut and Vertex Covermentioning

confidence: 99%

“…Our result for MAX CUT improves a result by de la Vega and Kenyon-Mathieu [6] who proved that the integrality gap remains 2 − ε after log c n rounds (where c < 1 is some fixed constant). Recently several very strong negative results were proved for other hierarchies: Schoenebeck, Trevisan, and Tulsiani [18] proved that the integrality gap for Vertex Cover and MAX CUT is 2 − ε after Ω(n) rounds of LS (Lovász-Schrijver relaxation of the LP program); Georgiou, Magen, Pitassi, and Tourlakis [7] proved that the integrality gap for Vertex Cover is 2 − ε after Ω( log n/ log log n) rounds of LS + (Lovász-Schrijver relaxation of the SDP program). Our results are not directly comparable with these results.…”

Section: Introductionmentioning

confidence: 99%

Integrality gaps for Sherali-Adams relaxations

Charikar

Makarychev

2009

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

140

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We prove strong lower bounds on integrality gaps of Sherali-Adams relaxations for MAX CUT, Vertex Cover, Sparsest Cut and other problems. Our constructions show gaps for Sherali-Adams relaxations that survive n δ rounds of lift and project. For MAX CUT and Vertex Cover, these show that even n δ rounds of Sherali-Adams do not yield a better than 2 − ε approximation.The main combinatorial challenge in constructing these gap examples is the construction of a fractional solution that is far from an integer solution, but yet admits consistent distributions of local solutions for all small subsets of variables. Satisfying this consistency requirement is one of the major hurdles to constructing Sherali-Adams gap examples. We present a modular recipe for achieving this, building on previous work on metrics with a local-global structure. We develop a conceptually simple geometric approach to constructing Sherali-Adams gap examples via constructions of consistent local SDP solutions.This geometric approach is surprisingly versatile. We construct Sherali-Adams gap examples for Unique Games based on our construction for MAX CUT together with a parallel repetition like procedure. This in turn allows us to obtain Sherali-Adams gap examples for any problem that has a Unique Games based hardness result (with some additional conditions on the reduction from Unique Games). Using this, we construct 2 − ε gap examples for Maximum Acyclic Subgraph that rules out any family of linear constraints with support at most n δ .

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“…This was later Downloaded 02/19/13 to 161.116.164.20. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php improved in [37,34,12] to more levels and to the semidefinite version LS+. For the SA hierarchy, it was shown in [8] that optimal gaps of 2 for vertex-cover and max-cut resist n Ω(1) levels.…”

Section: Related Workmentioning

confidence: 99%

Sherali-Adams relaxations and indistinguishability in counting logics

Atserias

Maneva

2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

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Abstract. Two graphs with adjacency matrices A and B are isomorphic if there exists a permutation matrix P for which the identity P T AP = B holds. Multiplying through by P and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorphism. We show that the levels of the Sherali-Adams (SA) hierarchy of linear programming relaxations applied to fractional isomorphism interleave in power with the levels of a well-known color-refinement heuristic for graph isomorphism called the Weisfeiler-Lehman algorithm, or, equivalently, with the levels of indistinguishability in a logic with counting quantifiers and a bounded number of variables. This tight connection has quite striking consequences. For example, it follows immediately from a deep result of Grohe in the context of logics with counting quantifiers that a fixed number of levels of SA suffice to determine isomorphism of planar and minor-free graphs. We also offer applications in both finite model theory and polyhedral combinatorics. First, we show that certain properties of graphs, such as that of having a flow circulation of a prescribed value, are definable in the infinitary logic with counting with a bounded number of variables. Second, we exploit a lower bound construction due to Cai, Fürer, and Immerman in the context of counting logics to give simple explicit instances that show that the SA relaxations of the vertex-cover and cut polytopes do not reach their integer hulls for up to Ω(n) levels, where n is the number of vertices in the graph.Key words. first-order logic, counting quantifiers, linear programming, Weisfeiler-Lehman algorithm, graph isomorphism, combinatorial optimization AMS subject classifications. 68Q17, 68Q19, 52B12, 05C60, 05C72, 03C80 DOI. 10.1137/120867834 1. Introduction. Let A and B be the adjacency matrices of two labeled graphs on {1, . . . , n}. The two graphs being isomorphic is equivalent to the existence of a permutation matrix P for which the relation P T AP = B holds. Multiplying both sides by P gives the equivalent condition AP = PB. At this point a linear programming relaxation suggests itself: relax the condition that P is a permutation matrix to a doubly stochastic matrix. How much coarser is this than actual isomorphism?The concept of fractional isomorphism as defined in the preceding paragraph falls within the framework of linear programming relaxations of combinatorial problems. Other types of relaxations of isomorphism include the color-refinement method called the Weisfeiler-Lehman (WL) algorithm. In this algorithm the vertices of the graphs are classified according to their degree, then according to the multiset of degrees of their neighbors, and so on until a fixed point is achieved. If the two graphs get partitions with different parameters, the graphs are definitely not isomorphic. As it turns out, fractional isomorphism and color refinement yield one and the same relaxation: it was shown by Ramana, Scheinerman, and Ullman [31] that two graphs

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Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut

Cited by 51 publications

References 9 publications

Linear Level Lasserre Lower Bounds for Certain k-CSPs

Linear Level Lasserre Lower Bounds for Certain k-CSPs

Integrality gaps for Sherali-Adams relaxations

Sherali-Adams relaxations and indistinguishability in counting logics

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