CSP gaps and reductions in the lasserre hierarchy

Tulsiani, Madhur

doi:10.1145/1536414.1536457

Cited by 110 publications

(116 citation statements)

References 33 publications

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“…This situation is similar to the very strong SDP gaps known for problems such as 3-XOR (see [14], [17]) for which deciding complete satisfiability is easy.…”

Section: Our Resultsmentioning

confidence: 52%

SDP Gaps for 2-to-1 and Other Label-Cover Variants

Guruswami

Khot

O’Donnell

et al. 2010

Automata, Languages and Programming

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Abstract. In this paper we present semidefinite programming (SDP) gap instances for the following variants of the Label-Cover problem, closely related to the Unique Games Conjecture: (i) 2-to-1 Label-Cover; (ii) 2-to-2 Label-Cover; (iii) α-constraint Label-Cover. All of our gap instances have perfect SDP solutions. For alphabet size K, the integral optimal solutions have value:Prior to this work, there were no known SDP gap instances for any of these problems with perfect SDP value and integral optimum tending to 0.

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“…This situation is similar to the very strong SDP gaps known for problems such as 3-XOR (see [14], [17]) for which deciding complete satisfiability is easy.…”

Section: Our Resultsmentioning

confidence: 52%

SDP Gaps for 2-to-1 and Other Label-Cover Variants

Guruswami

Khot

O’Donnell

et al. 2010

Automata, Languages and Programming

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“…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. For vertex-cover, a gap of 7/6 resists Ω(n) levels of Lasserre and hence of SA [33], and a gap of 1.36 resists n Ω(1) levels of Lasserre [38]. For max-cut, we could not find any published lower bound on the SA rank, but Schoenebeck informs us that his methods would yield a nontrivial gap for up to Ω(n) levels of Lasserre and hence SA.…”

Section: Related Workmentioning

confidence: 85%

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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“…To prove the Lasserre integrality gaps, we reduce from an integrality gap of random Max K-CSP from [Tul09]. The reduction and proof follow closely from those in [BCV + 12], in which similar integrality gaps for Densest k-Subgraph were shown.…”

Section: Lasserre Sdp Relaxation Of Label Cover Of Sizementioning

confidence: 92%

Approximation Algorithms for Label Cover and The Log-Density Threshold

Chlamtáč¹,

Manurangsi²,

Moshkovitz³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Many known optimal NP-hardness of approximation results are reductions from a problem called LabelCover. The input is a bipartite graph G = (L, R, E) and each edge e = (x, y) ∈ E carries a projection π e that maps labels to x to labels to y. The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio efficiently achievable for Label-Cover is of the form N −c where N = nk, n is the number of vertices, k is the number of labels, and 0 < c < 1 is some constant.Inspired by a framework originally developed for Densest k-Subgraph, we propose a "log density threshold" for the approximability of Label-Cover. Specifically, we suggest the possibility that the Label-Cover approximation problem undergoes a computational phase transition at the same threshold at which local algorithms for its random counterpart fail. This threshold is N 3−2 √ 2 ≈ N −0.17 . We then design, for any ε > 0, a polynomial-time approximation algorithm for semirandom Label-Cover whose approximation ratio is N 3−2 √ 2+ε . In our semi-random model, the input graph is random (or even just expanding), and the projections on the edges are arbitrary. time algorithm whose approximation ratio is roughly N −0.233 . The previous best efficient approximation ratio was N −0.25 . We present some evidence towards an N −c threshold by constructing integrality gaps for N Ω(1) rounds of the Sum-of-squares/Lasserre hierarchy of the natural relaxation of Label Cover. For general 2CSP the "log density threshold" is N −0.25 , and we give a polynomial-time algorithm in the semi-random model whose approximation ratio is N −0.25+ε for any ε > 0.

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CSP gaps and reductions in the lasserre hierarchy

Cited by 110 publications

References 33 publications

SDP Gaps for 2-to-1 and Other Label-Cover Variants

SDP Gaps for 2-to-1 and Other Label-Cover Variants

Sherali-Adams relaxations and indistinguishability in counting logics

Approximation Algorithms for Label Cover and The Log-Density Threshold

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