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A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
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 43 (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...
Tree pattern matching and subset matching in deterministic O(n log3 n)time
, 1999
"... Tree pattern matching and subset matching in deterministic O(n log ..."
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Cited by 37 (5 self)
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Tree pattern matching and subset matching in deterministic O(n log
Approximation Algorithms for Hitting Objects with Straight Lines
 Discrete Applied Mathematics
, 1989
"... In the hitting set problem one is given subsets of a finite set N and one has to find an X ae N of minimum cardinality that "hits" (intersects) all of them. The problem is NPhard. It is not known whether there exists a polynomialtime approximation algorithm for the hitting set prob ..."
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Cited by 31 (1 self)
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In the hitting set problem one is given subsets of a finite set N and one has to find an X ae N of minimum cardinality that "hits" (intersects) all of them. The problem is NPhard. It is not known whether there exists a polynomialtime approximation algorithm for the hitting set problem with a finite performance ratio. Special cases of the hitting set problem are described for which finite performance ratios are guaranteed. These problems arise in a geometric setting. We consider special cases of the following problem: Given n compact subsets of , find a set of straight lines of minimum cardinality so that each of the given subsets is hit by at least one line. The algorithms are based on several techniques of representing objects bypoints, not necessarily points on the objects, and solving (in some cases, only approximately) the problem of hitting the representative points. Finite performance ratios are obtained when the dimension, the number of types of sets to be hit and the number of directions of the hitting lines are bounded.
εApproximations with Minimum Packing Constraint Violation (Extended Abstract)
"... We present efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NPhard discrete optimization problem. Without any constraint violation, the εapproximation problem for ma ..."
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Cited by 8 (0 self)
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We present efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NPhard discrete optimization problem. Without any constraint violation, the εapproximation problem for many problems of this type is itself NPhard. Our methods provide polynomialtime εapproximations while attempting to minimize the packing constraint violation. Our methods
Nearly Optimal Vector Quantization via Linear Programming (Extended Abstract)
 In Proceedings of the IEEE Data Compression Conference
, 1992
"... We present new vector quantization algorithms based on the theory devel oped in [LiV]. The new approach is to formulate a vector quantization problem as a 01 integer linear program. We first solve its relaxed linear program by linear programming techniques. Then we transform the linear program ..."
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Cited by 3 (2 self)
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We present new vector quantization algorithms based on the theory devel oped in [LiV]. The new approach is to formulate a vector quantization problem as a 01 integer linear program. We first solve its relaxed linear program by linear programming techniques. Then we transform the linear program solution into a provably good solution for the vector quantization problem. These methods lead to the first known polynomialtime fullsearch vector quantization codebook design algorithm and tree pruning algorithm with provable worstcase performance guarantees. We also introduce the notion of pseudoraw, dom prued treestructured vector quatizers. Initial experimental results on image compression are very encouraging.
Around the PCP Theorem
 School of Computer Science, McGill University
, 1997
"... ing the situation, assume only that on input x the verifier V uses poly(logn) bits from a proof \Pi, uses O(log n) random bits, and is nonadaptive. We design a verifier V new which expects a proof \Pi new consisting of poly(n) entries each of poly(logn) bits. V new will read O(1) of these entries. ..."
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Cited by 2 (0 self)
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ing the situation, assume only that on input x the verifier V uses poly(logn) bits from a proof \Pi, uses O(log n) random bits, and is nonadaptive. We design a verifier V new which expects a proof \Pi new consisting of poly(n) entries each of poly(logn) bits. V new will read O(1) of these entries. The main idea is that \Pi new is supposed to be an encoding (using a lowdegree polynomial) of a proof \Pi which the old verifier V would have accepted with probability 1. The verifier checks using the low degree test that \Pi new is close to a lowdegree polynomial, and gets k values from it by the method in Section 3.2, where k is the polylog number of values demanded of \Pi by V . That is, \Pi new contains whatever information is needed for the lowdegree test, together with the coefficients of the line polynomial ~ \Pi(c(t)) for each polynomial c derived from a random point z k+1 . So V new reads one of these line polynomials (one "piece" of information), looks up f at O(1) points to ...