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Vertex Cover Might be Hard to Approximate to within 2  ɛ
"... Based on a conjecture regarding the power of unique 2prover1round games presented in [Khot02], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on kuniform hypergraph ..."
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Cited by 150 (11 self)
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Based on a conjecture regarding the power of unique 2prover1round games presented in [Khot02], we show that vertex cover is hard to approximate within any constant factor better than 2. We actually show a stronger result, namely, based on the same conjecture, vertex cover on kuniform hypergraphs is hard to approximate within any constant factor better than k.
The Importance of Being Biased
, 2002
"... The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 7/6 factor. ..."
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Cited by 88 (7 self)
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The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 7/6 factor.
Approximations of Weighted Independent Set and Hereditary Subset Problems
 JOURNAL OF GRAPH ALGORITHMS AND APPLICATIONS
, 2000
"... The focus of this study is to clarify the approximability of weighted versions of the maximum independent set problem. In particular, we report improved performance ratios in boundeddegree graphs, inductive graphs, and general graphs, as well as for the unweighted problem in sparse graphs. Wher ..."
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Cited by 71 (6 self)
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The focus of this study is to clarify the approximability of weighted versions of the maximum independent set problem. In particular, we report improved performance ratios in boundeddegree graphs, inductive graphs, and general graphs, as well as for the unweighted problem in sparse graphs. Where possible, the techniques are applied to related hereditary subgraph and subset problem, obtaining ratios better than previously reported for e.g. Weighted Set Packing, Longest Common Subsequence, and Independent Set in hypergraphs.
A better approximation ratio for the vertex cover problem
, 2005
"... We reduce the approximation factor for Vertex Cover to 2 − Θ ( 1 √ log n) (instead of the previous log log n 2 − Θ ( log n), obtained by BarYehuda and Even [2], and by Monien and Speckenmeyer [10]). The improvement of the vanishing factor comes as an application of the recent results of Arora, Rao, ..."
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Cited by 65 (0 self)
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We reduce the approximation factor for Vertex Cover to 2 − Θ ( 1 √ log n) (instead of the previous log log n 2 − Θ ( log n), obtained by BarYehuda and Even [2], and by Monien and Speckenmeyer [10]). The improvement of the vanishing factor comes as an application of the recent results of Arora, Rao, and Vazirani [1] that improved the approximation factor of the sparsest cut and balanced cut problems. In particular, we use the existence of two big and wellseparated sets of nodes in the solution of the semidefinite relaxation for balanced cut, proven in [1]. We observe that a solution of the semidefinite relaxation for vertex cover, when strengthened with the triangle inequalities, can be transformed into a solution of a balanced cut problem, and therefore the existence of big wellseparated sets in the sense of [1] translates into the existence of a big independent set. 1
Dependent rounding and its applications to approximation algorithms
 JOURNAL OF THE ACM
, 2006
"... We develop a new randomized rounding approach for fractional vectors defined on the edgesets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include: ffl low congestion multipath rout ..."
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Cited by 60 (7 self)
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We develop a new randomized rounding approach for fractional vectors defined on the edgesets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include: ffl low congestion multipath routing; ffl richer randomgraph models for graphs with a given degreesequence; ffl improved approximation algorithms for: (i) throughputmaximization in broadcast scheduling, (ii) delayminimization in broadcast scheduling, as well as (iii) capacitated vertex cover; and
A new multilayered PCP and the hardness of hypergraph vertex cover
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... Abstract Given a kuniform hypergraph, the EkVertexCover problem is to find the smallest subsetof vertices that intersects every hyperedge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that EkVertexCover is NPhard toapproximate within a ..."
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Cited by 55 (11 self)
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Abstract Given a kuniform hypergraph, the EkVertexCover problem is to find the smallest subsetof vertices that intersects every hyperedge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that EkVertexCover is NPhard toapproximate within a factor of ( k 1 &quot;) for arbitrary constants &quot;> 0 and k> = 3. The resultis nearly tight as this problem can be easily approximated within factor k. Our constructionmakes use of the biased LongCode and is analyzed using combinatorial properties of swise tintersecting families of subsets.We also give a different proof that shows an inapproximability factor of b k 2 c &quot;. In additionto being simpler, this proof also works for superconstant values of k up to (log N)1/c where
Parameterized complexity of generalized vertex cover problems
 In Proc. 9th WADS, volume 3608 of LNCS
, 2005
"... Abstract. Important generalizations of the Vertex Cover problem ..."
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Cited by 27 (3 self)
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Abstract. Important generalizations of the Vertex Cover problem
Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives
 In Proceedings of 52nd Annual Symposium on Foundations of Computer Science (FOCS
, 2011
"... We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and ..."
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Cited by 24 (3 self)
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We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NPhardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for lowrank approximations of a matrix in Frobenius norm using columns of the matrix. For all the above graph problems, we give an algorithm running in time nO(r/ε 2) with approximation ratio 1+εmin{1,λr} , where λr is the r’th smallest eigenvalue of the normalized graph Laplacian L. In the case of graph bisection and small set expansion, the number of vertices in the cut is within lowerorder terms of the stipulated bound. Our results imply (1 + O(ε)) factor approximation in time nO(r