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Proof verification and hardness of approximation problems
 IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI
, 1992
"... We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probabilit ..."
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Cited by 797 (39 self)
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We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof " with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [6] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). As a consequence we prove that no MAX SNPhard problem has a polynomial time approximation scheme, unless NP=P. The class MAX SNP was defined by Papadimitriou and Yannakakis [82] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating the maximum clique size in an Nvertex graph to within a factor of N ɛ is NPhard.
Hardness Of Approximations
, 1996
"... This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems. ..."
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Cited by 117 (5 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
, 1992
"... The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and acce ..."
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Cited by 65 (8 self)
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The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and accepting probabilistic guarantees from the verifier [BFL91, BFLS91, FGL + 91, AS92]. We improve upon the efficiency of the proof systems developed above and obtain proofs which can be verified probabilistically by examining only a constant number of (randomly chosen) bits of the proof. The efficiently verifiable proofs constructed here rely on the structural properties of lowdegree polynomials. We explore the properties of these functions by examining some simple and basic questions about them. We consider questions of the form: • (testing) Given an oracle for a function f, is f close to a lowdegree polynomial? • (correcting) Let f be close to a lowdegree polynomial g, is it possible to efficiently reconstruct the value of g on any given input using an oracle for f? 2 The questions described above have been raised before in the context of coding theory as the problems of errordetecting and errorcorrecting of codes. More recently
The approximability of NPhard problems
 In Proceedings of the Annual ACM Symposium on Theory of Computing
, 1998
"... Many problems in combinatorial optimization are NPhard (see [60]). This has forced researchers to explore techniques for dealing with NPcompleteness. Some have considered algorithms that solve “typical” ..."
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Cited by 17 (0 self)
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Many problems in combinatorial optimization are NPhard (see [60]). This has forced researchers to explore techniques for dealing with NPcompleteness. Some have considered algorithms that solve “typical”
zapproximations
 Journal of Algorithms
, 2001
"... Approximation algorithms for NPhard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ..."
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Cited by 17 (3 self)
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Approximation algorithms for NPhard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ratio may not be very meaningful for example, if the ratio of the worst solution to the best solution is at most some constant ff, then an approximation algorithm with factor ff may in fact yield the worst solution! To overcome this hurdle (among others), several authors have independently suggested the use of a different measure which we call zapproximation. An algorithm is an ff zapproximation if it runs in polynomial time, and produces a solution whose distance from the optimal one is at most ff times the distance between the optimal solution and the worst possible solution. The results known so far about zapproximations are either of the inapproximability type or rather straightforward observations. We design polynomial time algorithms for several fundamental discrete optimization problems, in particular we obtain a zapproximation factor of 1 2 for the directed traveling salesman problem (TSP) (with no triangle inequality assumption). For the undirected TSP this improves to
Differential approximation results for the traveling salesman problem with distances 1 and 2
 Information Processing Letters
, 2001
"... We prove that both minimum and maximum traveling salesman problems on complete graphs with edgedistances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman w ..."
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Cited by 14 (6 self)
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We prove that both minimum and maximum traveling salesman problems on complete graphs with edgedistances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any ɛ> 0, it is NPhard to approximate both problems better than within 741/742 + ɛ. The same results hold when dealing with a generalization of min and max TSP12, where instead of 1 and 2, edges are valued by a and b.
On the differential approximation of min set cover
 Theor. Comput. Sci
, 2005
"... We present in this paper differential approximation results for min set cover and min weighted set cover. We first show that the differential approximation ratio of the natural greedy algorithm for min set cover is bounded below by 1.365/ ∆ and above by 4/(∆+1), where ∆ is the maximum setcardinalit ..."
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Cited by 2 (2 self)
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We present in this paper differential approximation results for min set cover and min weighted set cover. We first show that the differential approximation ratio of the natural greedy algorithm for min set cover is bounded below by 1.365/ ∆ and above by 4/(∆+1), where ∆ is the maximum setcardinality in the min set coverinstance. Next we study another approximation algorithm for min set cover that computes 2optimal solutions, i.e., solutions that cannot be improved by removing two sets belonging to them and adding another set not belonging to them. We prove that the differential approximation ratio of this second algorithm is bounded below by 2/( ∆ + 1) and that this bound is tight. Finally, we study an approximation algorithm for min weighted set cover and provide a tight lower bound of 1/∆. Our results identically hold for max hypergraph independent set in both the standard and the differential approximation paradigms. 1