Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling

Manokaran, Rajsekar; Naor, Joseph; Raghavendra, Prasad; Schwartz, Roy

doi:10.1145/1374376.1374379

Cited by 55 publications

(52 citation statements)

References 23 publications

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“…However, assuming Khot's Unique Games Conjecture (UGC) [12], an essentially optimal k − ε hardness of approximating the minimum vertex cover on k-uniform hypergraphs was shown by Khot and Regev [14]. In more recent works the UGC has been used to relate the inapproximability of various classes of constraint satisfaction problems (CSPs) to the corresponding semi-definite programming (SDP) integrality gap [19], or the linear programming (LP) integrality gap [17] [15]. The recent work of Kumar et al [15] generalizes the result of [14] and shall be of particular interest in this work.…”

Section: Introductionmentioning

confidence: 99%

On the Inapproximability of Vertex Cover on k-Partite k-Uniform Hypergraphs

Guruswami

Saket

2010

Automata, Languages and Programming

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Abstract. Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely finding a small vertex cover in kuniform k-partite hypergraphs, when the k-partition is given as input. For this problem Lovász [16] gave a k 2 factor LP rounding based approximation, and a matching k 2 − o(1) integrality gap instance was constructed by Aharoni et al. [1]. We prove the following results, which are the first strong hardness results for this problem (here ε > 0 is an arbitrary constant):-NP-hardness of approximating within a factor of k 4 − ε , and -Unique Games-hardness of approximating within a factor of k 2 − ε , showing optimality of Lovász's algorithm under the Unique Games conjecture. The NP-hardness result is based on a reduction from minimum vertex cover in r-uniform hypergraphs for which NP-hardness of approximating within r−1−ε was shown by Dinur et al. [5]. The Unique Games-hardness result is obtained by applying the recent results of Kumar et al. [15], with a slight modification, to the LP integrality gap due to Aharoni et al. [1]. The modification is to ensure that the reduction preserves the desired structural properties of the hypergraph.

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Section: Introductionmentioning

confidence: 99%

On the Inapproximability of Vertex Cover on k-Partite k-Uniform Hypergraphs

Guruswami

Saket

2010

Automata, Languages and Programming

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“…The geometric relaxation of Cȃlinescu et al [7] carries much importance, since Manokaran et al [28] proved that its integrality gap can be translated to a hardness result for the Multiway Cut problem with the exact same value, assuming the unique games conjecture. Hence, this suggests that the best possible approximation guarantee for the Multiway Cut problem can be obtained by rounding the geometric relaxation of Cȃlinescu et al [7].…”

Section: Related Workmentioning

confidence: 99%

Simplex Transformations and the Multiway Cut Problem

Buchbinder¹,

Schwartz²,

Weizman³

2017

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

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We consider Multiway Cut, a basic graph partitioning problem in which the goal is to find the minimum weight collection of edges disconnecting a given set of special vertices called terminals. Multiway Cut admits a well known simplex embedding relaxation, where rounding this embedding is equivalent to partitioning the simplex. Current best known solutions to the problem are comprised of a mix of several different ingredients, resulting in intricate algorithms. Moreover, the best of these algorithms is too complex to fully analyze analytically and its approximation factor was verified using a computer. We propose a new approach to simplex partitioning and the Multiway Cut problem based on general transformations of the simplex that allow dependencies between the different variables. Our approach admits much simpler algorithms, and in addition yields an approximation guarantee for the Multiway Cut problem that (roughly) matches the current best computer verified approximation factor.

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“…Partitioning problems of this type have been investigated recently in [2,3,6,5]. The main result of [2] is a 2-approximation for Submodular Multiway Partition, a special case of Submodular Labeling with Color Lists where the color lists are either singletons ("terminals") or equal to [k] (unrestricted).…”

Section: Introductionmentioning

confidence: 99%

“…This generalizes a 2-approximation for Uniform Metric Labeling [5] which corresponds to the  = 2 case. On the hardness side, the strongest negative result was a hardness of (2 -)-approximation assuming the Unique Games Conjecture [6].…”

Section: Introductionmentioning

confidence: 99%