Improved Approximation Algorithms for Label Cover Problems

Charikar, Moses; Hajiaghayi, MohammadTaghi; Karloff, Howard

doi:10.1007/s00453-010-9464-3

Cited by 24 publications

(14 citation statements)

References 17 publications

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“…Approximation algorithms for projection games were researched, and the conjecture is consistent with the state of the art algorithm, giving 1/ε = O( 3 √ N k) [CHK09] (Note that the formulation in [CHK09] is slightly different than ours -they have a vertex per pair (vertex, assignment) in our formulation).…”

Section: Previous Worksupporting

confidence: 53%

The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover

Moshkovitz

2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

134

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We suggest the research agenda of establishing new hardness of approximation results based on the "projection games conjecture", i.e., an instantiation of the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell to projection games.We pursue this line of research by establishing a tight N P-hardness result for the SetCover problem. Specifically, we show that under the projection games conjecture (in fact, under a quantitative version of the conjecture that is only slightly beyond the reach of current techniques), it is N P-hard to approximate Set-Cover on instances of size N to within (1 − α) ln N for arbitrarily small α > 0. Our reduction establishes a tight trade-off between the approximation accuracy α and the time required for the approximation 2

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Section: Previous Worksupporting

confidence: 53%

The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover

Moshkovitz

2012

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

134

View full text Add to dashboard Cite

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“…This gives a approximation ratio of N 1/4+ε for semirandom instances. For worst case 2CSP the best efficient algorithm known gives only Θ(N 1/3 )-approximation [CHK09], but works in the more general case where the fraction of satisfied edges in the completeness case is arbitrary.…”

Section: Additional Resultsmentioning

confidence: 99%

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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“…DSF is also known [32] to be as hard as MAX-REP, a basic problem used for hardness reductions, for which the current best approximation ratio is O (n 1/3 ) [33].…”

Section: Directed Steiner Forestmentioning

confidence: 99%

Approximation algorithms for spanner problems and Directed Steiner Forest

Berman

Bhattacharyya

Makarychev

et al. 2013

Information and Computation

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We present an O ( √ n log n)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V , E) with nonnegative edge lengths d : E → R 0 and a stretch k 1,The previous best approximation ratio wasÕ (n 2/3 ), due to Dinitz and Krauthgamer (STOC '11).We also improve the approximation ratio for the important special case of directed 3-spanners with unit edge lengths fromÕ ( √ n ) to O (n 1/3 log n). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS '10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and Krauthgamer's lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly implies an O (n 1/3 log n)-approximation for the 3-spanner problem on undirected graphs with unit lengths. An easy O ( √ n )-approximation algorithm for this problem has been the best known for decades. Finally, we consider the Directed Steiner Forest problem: given a directed graph with edge costs and a collection of ordered vertex pairs, find a minimum-cost subgraph that contains a path between every prescribed pair. We obtain an approximation ratio of O (n 2/3+ ) for any constant > 0, which improves the O (n · min(n 4/5 , m 2/3 )) ratio due to Feldman, Kortsarz and Nutov (JCSS'12).

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Improved Approximation Algorithms for Label Cover Problems

Abstract: In this paper we consider both the maximization variant Max Rep and the minimization variant Min Rep of the famous Label Cover problem, for which, till now, the best approximation ratios known were

Cited by 24 publications

References 17 publications

The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover

The Projection Games Conjecture and the NP-Hardness of ln n-Approximating Set-Cover

Approximation Algorithms for Label Cover and The Log-Density Threshold

Approximation algorithms for spanner problems and Directed Steiner Forest

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