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16
Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs
"... We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Gameshard to approximate log log d to within a factor 2−(2+od(1)). This exactly log d matches the algorithmic result of Halperin [1] up to the od(1) term. • Independent Set ..."
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We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Gameshard to approximate log log d to within a factor 2−(2+od(1)). This exactly log d matches the algorithmic result of Halperin [1] up to the od(1) term. • Independent Set is Unique Gameshard to approxid mate to within a factor O( log2). This improves the d d logO(1) Unique Games hardness result of Samorod
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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Cited by 15 (1 self)
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
Minimizing the sum of weighted completion times in a concurrent open shop
 OPERATIONS RESEARCH LETTERS
, 2010
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Bypassing UGC from some Optimal Geometric Inapproximability Results
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 177
, 2010
"... The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these ..."
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The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this work we bypass the UGC assumption in inapproximability results for two geometric problems, obtaining a tight NPhardness result in each case. The first problem known as the Lp Subspace Approximation is a generalization of the classic least squares regression problem. Here, the input consists of a set of points S = {a1,..., am} ⊆ R n and a parameter k (possibly depending on n). The goal is to find a subspace H of R n of dimension k that minimizes the sum of the p th powers of the distances to the points. For p = 2, k = n − 1, this reduces to the least squares regression problem, while for p = ∞, k = 0 it reduces to the problem of finding a ball of minimum radius enclosing all the points. We show that for any fixed p (2 < p < ∞) it is NPhard to approximate this problem to within a factor of γp − ɛ for constant ɛ> 0, where γp is the pth moment of a standard Gaussian variable. This matches the factor γp approximation algorithm obtained by Deshpande, Tulsiani and Vishnoi
On the Inapproximability of Vertex Cover on kPartite kUniform Hypergraphs
"... Computing a minimum vertex cover in graphs and hypergraphs is a wellstudied 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, ..."
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Computing a minimum vertex cover in graphs and hypergraphs is a wellstudied 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 kpartite hypergraphs, when the kpartition 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): – NPhardness of approximating within a factor of ( k 4 − ε) , and – Unique Gameshardness of approximating within a factor of ( k 2 − ε), showing optimality of Lovász’s algorithm under the Unique Games conjecture. The NPhardness result is based on a reduction from minimum vertex cover in runiform hypergraphs for which NPhardness of approximating within r−1−ε was shown by Dinur et al. [5]. The Unique Gameshardness 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.
Hardness of finding independent sets in 2colorable and almost 2colorable hypergraphs
, 2014
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Flexible Coloring
"... Motivated by reliability considerations in data deduplication for storage systems, we introduce the problem of flexible coloring. Given a hypergraph H and the number of allowable colors k, a flexible coloring of H is an assignment of one or more colors to each vertex such that, for each hyperedge, i ..."
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Motivated by reliability considerations in data deduplication for storage systems, we introduce the problem of flexible coloring. Given a hypergraph H and the number of allowable colors k, a flexible coloring of H is an assignment of one or more colors to each vertex such that, for each hyperedge, it is possible to choose a color from each vertex’s color list so that this hyperedge is strongly colored (i.e., each vertex has a different color). Different colors for the same vertex can be chosen for different incident hyperedges (hence the term flexible). The goal is to minimize color consumption, namely, the total number of colors assigned, counting multiplicities. Flexible coloring is NPhard and trivially s − (s−1)k n approximable, where s is the size of the largest hyperedge, and n is the number of vertices. Using a recent result by Bansal and Khot, we show that if k is constant, then it is UGChard to approximate to within a factor of s − ε, for arbitrarily small constant ε> 0. s − (s−1)k k ′ Lastly, we present an algorithm with an approximation ratio, where k ′ is number of colors used by a strong coloring algorithm for H. Keywords: graph coloring, hardness of approximation
New Results in the Theory of Approximation  Fast Graph Algorithms and Inapproximability
, 2013
"... For several basic optimization problems, it is NPhard to find an exact solution. As a result, understanding the best possible tradeoff between the running time of an algorithm and its approximation guarantee, is a fundamental question in theoretical computer science, and the central goal of the th ..."
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For several basic optimization problems, it is NPhard to find an exact solution. As a result, understanding the best possible tradeoff between the running time of an algorithm and its approximation guarantee, is a fundamental question in theoretical computer science, and the central goal of the theory of approximation. There are two aspects to the theory of approximation: (1) efficient approximation algorithms that establish tradeoffs between approximation guarantee and running time, and (2) inapproximability results that give evidence against them. In this thesis, we contribute to both facets of the theory of approximation. In the first part of this thesis, we present the first nearlineartime algorithm for Balanced Separator given a graph, partition its vertices into two roughly equal parts without cutting too many edges that achieves the best approximation guarantee possible for algorithms in its class. This is a classic graph partitioning problem and has deep connections to several areas of both theory and practice, such as metric embeddings, Markov chains, clustering, etc.
Tight Approximation Bounds for Vertex Cover on Dense kPartite Hypergraphs
"... We establish almost tight upper and lower approximation bounds for the Vertex Cover problem on dense kpartite hypergraphs. ..."
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We establish almost tight upper and lower approximation bounds for the Vertex Cover problem on dense kpartite hypergraphs.
New and Improved Bounds for the Minimum Set Cover Problem
"... Abstract. We study the relationship between the approximation factor for the SetCover problem and the parameters ∆ : the maximum cardinality of any subset, and k: the maximum number of subsets containing any element of the ground set. We show an LP rounding based approxi− ln ∆ mation of (k − 1)(1 ..."
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Abstract. We study the relationship between the approximation factor for the SetCover problem and the parameters ∆ : the maximum cardinality of any subset, and k: the maximum number of subsets containing any element of the ground set. We show an LP rounding based approxi− ln ∆ mation of (k − 1)(1 − e k−1) + 1, which is substantially better than the classical algorithms in the range k ≈ ln ∆, and also improves on related previous works [19, 22]. For the interesting case when k = θ(log ∆) we also exhibit an integrality gap which essentially matches our approximation ( algorithm.) We also prove a hardness of approximation factor of Ω when k = θ(log ∆). This is the first study of the hardness log ∆ (log log ∆) 2 factor specifically for this range of k and ∆, and improves on the only other such result implicitly proved in [18]. 1