On the hardness of approximating vertex cover

Dinur, Irit; Safra, Shmuel

doi:10.4007/annals.2005.162.439

Cited by 554 publications

(382 citation statements)

References 47 publications

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“…There is an almost trivial k-approximation by finding a maximal matching and the best known algorithms achieve only a slight improvement of (1 − o(1))k [9,13]. On the inapproximability side, for the case k = 2, Dinur and Safra [5] obtained 1.36 NP-hardness improving on the 7 6 − ε hardness by Håstad [10]. For larger k, a sequence of works obtained hardness factor k 1/19 (Trevisan [22]), 2 − ε for some constant k (Goldreich [8]), 2 − ε for k = 4, 3 2 − ε for k = 3, k 1−o(1) (Holmerin [12,11]), k − 3 − ε (Dinur et al [3]), and finally k − 1 − ε (Dinur et al [4]) that superseded all earlier results for k ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

Bansal

Khot

2010

Automata, Languages and Programming

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Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems. We obtain a hardness of 2 − ε for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2 − ε for minimizing the makespan in the assembly line problem.These results follow from a new inapproximability result for the Vertex Cover problem on k-uniform hypergraphs that is stronger and simpler than previous results. We show that assuming the UGC, for every k ≥ 2, the problem is inapproximable within k − ε even when the hypergraph is almost k-partite.

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Section: Introductionmentioning

confidence: 99%

Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

Bansal

Khot

2010

Automata, Languages and Programming

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“…Combining Lemma 1 with the fact that the vertex cover problem is NP-hard to approximate within a factor of 1.36 [4] immediately gives us the following:…”

Section: Fig 3 Reduction From Vertex Covermentioning

confidence: 99%

Connections in Networks: Hardness of Feasibility Versus Optimality

Conrad

Gomes

Hoeve

et al. 2007

Lecture Notes in Computer Science

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Abstract. We study the complexity of combinatorial problems that consist of competing infeasibility and optimization components. In particular, we investigate the complexity of the connection subgraph problem, which occurs, e.g., in resource environment economics and social networks. We present results on its worst-case hardness and approximability. We then provide a typical-case analysis by means of a detailed computational study. First, we identify an easy-hard-easy pattern, coinciding with the feasibility phase transition of the problem. Second, our experimental results reveal an interesting interplay between feasibility and optimization. They surprisingly show that proving optimality of the solution of the feasible instances can be substantially easier than proving infeasibility of the infeasible instances in a computationally hard region of the problem space. We also observe an intriguing easy-hard-easy profile for the optimization component itself.

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“…(These conjectures will be described formally in Section 3.) An instance of Label-Cover with α-constraints was also implicit in the result of Dinur and Safra [4] on the hardness of approximating minimum vertex cover.…”

Section: The Unique Games Conjecture and Its Variantsmentioning

confidence: 99%

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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On the hardness of approximating vertex cover

Abstract: We prove the Minimum Vertex Cover problem to be NP-hard to approximate to within a factor of 1.3606, extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and to borrow and extend ideas from several fields.

Cited by 554 publications

References 47 publications

Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems

Connections in Networks: Hardness of Feasibility Versus Optimality

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

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