We show that vertex cover is hard to approximate within any constant factor better than 2 where the hardness is based on a conjecture regarding the power of unique 2-prover-1-round games presented in [15]. We actually show a stronger result, namely, based on the same conjecture, vertex cover on k-uniform hypergraphs is hard to approximate within any constant factor better than k.

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 NP-hard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1

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 bounded-degree 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.

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Abstract Given a k-uniform hypergraph, the Ek-Vertex-Cover 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 Ek-Vertex-Cover is NP-hard toapproximate within a factor of ( k- 1- ") for arbitrary constants "> 0 and k> = 3. The resultis nearly tight as this problem can be easily approximated within factor k. Our constructionmakes use of the biased Long-Code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets.We also give a different proof that shows an inapproximability factor of b k 2 c- ". In additionto being simpler, this proof also works for super-constant values of k up to (log N)1/c where

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Abstract We develop a new randomized rounding approach for fractional vectors defined on the edge-sets 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 multi-path routing; ffl richer random-graph models for graphs with a given degree-sequence; ffl improved approximation algorithms for: (i) throughput-maximization in broadcast scheduling, (ii) delay-minimization in broadcast scheduling, as well as (iii) capacitated vertex cover; and

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 Bar-Yehuda 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 well-separated 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 well-separated sets in the sense of [1] translates into the existence of a big independent set. 1

Abstract. Important generalizations of the Vertex Cover problem

In this paper we study the capacitated vertex cover problem, a generalization of the well-known vertex cover problem. Given a graph G = (V, E) with weights on the vertices, the goal is to cover all the edges by picking a cover of minimum weight from the vertices. When we pick a copy of a vertex, we pay the weight of the vertex and cover up to a pre-specified number of edges incident on this vertex (its capacity). The problem is NP-hard. We give a primal–dual based approximation algorithm with an approximation guarantee of 2, and study several generalizations, as well as the problem restricted to trees.