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33
On the power of unique 2prover 1round games
 In Proceedings of the 34th Annual ACM Symposium on Theory of Computing
, 2002
"... ABSTRACT A 2prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2prover game is the maximum acceptance probability of the verifier over all the prover strategi ..."
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Cited by 233 (19 self)
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ABSTRACT A 2prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make the following conjecture regarding the power of unique 2prover games, which we call the Unique Games Conjecture: The Unique Games Conjecture: For arbitrarily small constants i; ffi? 0, there exists a constant k = k(i; ffi) such that it is NPhard to determine whether a unique 2prover game with answers from a domain of size k has value at least 1 \Gamma i or at most ffi. We show that a positive resolution of this conjecture would imply the following hardness results:
The Dense kSubgraph Problem
 Algorithmica
, 1999
"... This paper considers the problem of computing the dense kvertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense ksubgraph (D ..."
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Cited by 162 (9 self)
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This paper considers the problem of computing the dense kvertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ffi ), for some ffi ! 1=3. 1 Introduction We study the dense ksubgraph (DkS) maximization problem, of computing the dense k vertex subgraph of a given graph. That is, on input a graph G and a parameter k, we are interested in finding a set of k vertices with maximum average degree in the subgraph induced by this set. As this problem is NPhard (say, by reduction from Clique), we consider approximation algorithms for this problem. We obtain a polynomial time algorithm that on any input (G; k) returns a subgraph of size k whose average degree is within a factor of at most n ffi from the optimum solution, where n is the number of vertices in the input graph G, and ffi ! 1=3 is some universal constant. Unfortunately, we are unable to present a complementary negati...
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 96 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs
, 1999
"... We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a perf ..."
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Cited by 92 (6 self)
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We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most &Delta;, the algorithm achieves a performance ratio of 2  (1  o(1)) 2 ln ln \Delta ln \Delta for large \Delta, which improves the previously known ratio of 2 \Gamma log \Delta+O(1) \Delta obtained by Halldórsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For kuniform hypergraphs with n vertices, we achieve a ratio of k \Gamma (1 \Gamma o(1)) k ln ln n ln n for large n, and for kuniform hypergraphs with maximum degree at most \Delta, the algorithm achieves a ratio of k \Gamma (1 \Gamma o(1)) k(k\Gamma1) ln ln \Delta ln \Delta for large \Delta. These results considerably improve the previous best ratio of k(1\Gammac=\Delta 1 k\Gamma1 ) for bounded degree kuniform hypergraphs, and k(1 \Gamma c=n k\Gamma1 k ) for general kuniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldórsson.
Finding a large hidden clique in a random graph
, 1998
"... ABSTRACT: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph Gn,1�2 Ž., and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomia ..."
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Cited by 79 (5 self)
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ABSTRACT: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph Gn,1�2 Ž., and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k�cn0.5 ˇ, for
NonApproximability Results for Optimization Problems on Bounded Degree Instances
, 2001
"... We prove some nonapproximability results for restrictions of basic combinatorial optimization problems to instances of bounded \degree" or bounded \width." Speci cally: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approxima ..."
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Cited by 76 (5 self)
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We prove some nonapproximability results for restrictions of basic combinatorial optimization problems to instances of bounded \degree" or bounded \width." Speci cally: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor 7=8+O(1= B), unless RP = NP . Hastad [18] proved that the problem is approximable to within a factor 7=8+1=64B in polynomial time, and that is hard to approximate to within a factor 7=8 + 1=(log B) 3 . Our result uses a new randomized reduction from general instances of Max 3SAT to boundedoccurrences instances. The randomized reduction applies to other Max SNP problems as well.
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 53 (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.
PseudoRandom Graphs
 IN: MORE SETS, GRAPHS AND NUMBERS, BOLYAI SOCIETY MATHEMATICAL STUDIES 15
"... ..."
Randomized graph products, chromatic numbers, and the Lovász thetafunction
 Combinatorica
, 1996
"... For a graph G, let ff(G) denote the size of the largest independent set in G, and let #(G) denote the Lov'asz #function on G. We prove that for some c ? 0, there exists an infinite family of graphs such that #(G) ? ff(G)n=2 c p log n , where n denotes the number of vertices in a graph. ..."
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Cited by 41 (6 self)
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For a graph G, let ff(G) denote the size of the largest independent set in G, and let #(G) denote the Lov'asz #function on G. We prove that for some c ? 0, there exists an infinite family of graphs such that #(G) ? ff(G)n=2 c p log n , where n denotes the number of vertices in a graph. This disproves a known conjecture regarding the # function. As part of our proof, we analyse the behavior of the chromatic number in graphs under a randomized version of graph products. This analysis extends earlier work of Linial and Vazirani, and of Berman and Schnitger, and may be of independent interest. 1 Introduction Lov'asz [21] introduced the # function in order to study the so called "Shannon Capacity" of graphs. For every graph G, the # function enjoys the following sandwich property: ff(G) #(G) (G) where ff(G) is the size of the largest independent set in G, and (G) is the clique cover number of G ((G) = ( G), the chromatic number of the complement of G). This sandwich prop...