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17
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 102 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
The Complexity and Approximability of Finding Maximum Feasible Subsystems of Linear Relations
 Theoretical Computer Science
, 1993
"... We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of rela ..."
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Cited by 77 (12 self)
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We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of relations =, , ? and 6=. Various constrained versions of Max FLS, where a subset of relations must be satisfied or where the variables take bounded discrete values, are also considered. We establish the complexity of solving these problems optimally and, whenever they are intractable, we determine their degree of approximability. Max FLS with =, or ? relations is NPhard even when restricted to homogeneous systems with bipolar coefficients, whereas it can be solved in polynomial time for 6= relations with real coefficients. The various NPhard versions of Max FLS belong to different approximability classes depending on the type of relations and the additional constraints. We show that the ran...
Proof Checking and Approximation: Towards Tight Results
 SIGACT News
, 1996
"... Introduction The last few years have seen much progress in proving "nonapproximability results" for wellknown NPhard optimization problems. As we know, the breakthrough has come by the application of results from probabilistic proof checking. It is an area that seems to continue to su ..."
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Cited by 16 (0 self)
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Introduction The last few years have seen much progress in proving "nonapproximability results" for wellknown NPhard optimization problems. As we know, the breakthrough has come by the application of results from probabilistic proof checking. It is an area that seems to continue to surprise: since the connection was discovered in 1991 (Feige et. al. [21]), not only have nonapproximability results emerged for a wide range of problems, but the factors shown hard steadily increase. Today, tight results are known for central problems like MaxClique and MinSetCover. (That is, the approximation algorithms we have for these problems can be shown to be the best possible). Such results also seem to be in sight for ChromNum. These are remarkable things, especially in the light of our knowledge of just five years ago. And meanwhile we continue to make progress on the MaxSNP front, where both the algor
A Combinatorial Consistency Lemma with application to proving the PCP Theorem
 PRELIMINARY VERSION IN RANDOM
, 1997
"... The current proof of the PCP Theorem (i.e., NP = PCP(log; O(1))) is very complicated. One source of difficulty is the technically involved analysis of lowdegree tests. Here, we refer to the difficulty of obtaining strong results regarding lowdegree tests; namely, results of the type obtained and u ..."
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Cited by 16 (4 self)
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The current proof of the PCP Theorem (i.e., NP = PCP(log; O(1))) is very complicated. One source of difficulty is the technically involved analysis of lowdegree tests. Here, we refer to the difficulty of obtaining strong results regarding lowdegree tests; namely, results of the type obtained and used by Arora and Safra and Arora et. al. In this paper, we eliminate the need to obtain such strong results on lowdegree tests when proving the PCP Theorem. Although we do not get rid of lowdegree tests altogether, using our results it is now possible to prove the PCP Theorem using a simpler analysis of lowdegree tests (which yields weaker bounds). In other words, we replace the strong algebraic analysis of lowdegree tests presented by Arora and Safra and Arora et. al. by a combinatorial lemma (which does not refer to lowdegree tests or polynomials).
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 15 (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”
Approximation algorithms for MAX SAT: A better performance ratio at the cost of a longer running time
, 1998
"... We describe approximation algorithms for (unweighted) MAX SAT with performance ratios arbitrarily close to 1 (in particular, when performance ratios exceed the limit of polynomialtime approximation). Namely, we show how to construct an (# + #)approximation algorithm A from a given polynomialtime ..."
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Cited by 13 (8 self)
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We describe approximation algorithms for (unweighted) MAX SAT with performance ratios arbitrarily close to 1 (in particular, when performance ratios exceed the limit of polynomialtime approximation). Namely, we show how to construct an (# + #)approximation algorithm A from a given polynomialtime #approximation algorithm A 0 . The algorithm A runs in time of the order # #(1#) 1 K , where # is the golden ratio (# 1.618) and K is the number of clauses in the input formula. Thus we estimate the cost of improving a performance ratio. Similar constructions for MAX 2SAT and MAX 3SAT are described too. We apply our constructions to some known polynomialtime algorithms taken as A 0 and give upper bounds on the running time of the respective algorithms A. 1 Introduction In the MAX SAT problem we are given a Boolean formula represented by a set of clauses, and we seek a truth assignment that maximizes the number of satisfied clauses. An #approximation algorithm for MAX SAT i...
Fault Tolerant Circuits and Probabilistically Checkable Proofs
 IN PROCEEDINGS OF THE 10TH ANNUAL STRUCTURE IN COMPLEXITY THEORY
, 1995
"... We introduce a new model of fault tolerant Boolean circuits. We allow an adversary to choose some gates to be faulty, unlike the model considered by von Neumann and Pippenger where the errors are randomly distributed. Our model also differs from previous models that considered nonrandom faults. Our ..."
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Cited by 5 (1 self)
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We introduce a new model of fault tolerant Boolean circuits. We allow an adversary to choose some gates to be faulty, unlike the model considered by von Neumann and Pippenger where the errors are randomly distributed. Our model also differs from previous models that considered nonrandom faults. Our main result is that every symmetric function has a small (size O(n), depth O(logn)) fault tolerant circuit that will compute the function adequately, even if a small constant fraction of the gates is modified by an adversary. We also show a perhaps unexpected relation between our model and probabilistically checkable proofs.
MultiProver Encoding Schemes and ThreeProver Proof Systems
, 1994
"... Suppose two provers agree in a polynomial p and want to reveal a single value y = p(x) to a verifier where x is chosen arbitrarily by the verifier. Whereas honest provers should be able to agree on any polynomial p the verifier wants to be sure that with any (cheating) pair of provers the value y he ..."
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Cited by 2 (0 self)
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Suppose two provers agree in a polynomial p and want to reveal a single value y = p(x) to a verifier where x is chosen arbitrarily by the verifier. Whereas honest provers should be able to agree on any polynomial p the verifier wants to be sure that with any (cheating) pair of provers the value y he receives is a polynomial function of x. We formalize this question and introduce multiprover (quasi)encoding schemes to solve it.