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Probabilistic checking of proofs: a new characterization of NP
 Journal of the ACM
, 1998
"... Abstract. We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from ..."
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Cited by 368 (28 self)
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Abstract. We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof. We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NPhard.
Efficient probabilistically checkable proofs and applications to approximation
 In Proceedings of STOC93
, 1993
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On the Hardness of Approximating MAX kCUT and its Dual
, 1995
"... . We study the Max kCut problem and its dual, the Min kPartition problem: given G = (V; E) and w : E ! R + , find an edge set of minimum weight whose removal makes G kcolorable. For the Max kCut problem we show that, if P 6= NP, no polynomial time approximation algorithm can achieve a relativ ..."
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Cited by 27 (2 self)
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. We study the Max kCut problem and its dual, the Min kPartition problem: given G = (V; E) and w : E ! R + , find an edge set of minimum weight whose removal makes G kcolorable. For the Max kCut problem we show that, if P 6= NP, no polynomial time approximation algorithm can achieve a relative error of 1=oek, where oe = 132. This should be compared to the wellknown fact that a naive randomized heuristics delivers approximations whose relative error is 1=k. For the Min kPartition problem, we show that for k ? 2 and for every ffl ? 0, there exists a constant ff such that the problem cannot be approximated within ffjV j 2\Gammaffl , even for dense graphs. Both problems are directly related to the frequency allocation problem for cellular (mobile) telephones, an industrial application of increasing relevance. 1 Introduction A good starting point to motivate the problems studied in this paper is to consider the frequency allocation problem for cellular telephones. We are given a...
A Well Characterized Approximation Problem
 In Information processing letters, Vol 47:6
, 1993
"... We consider the following NP optimization problem: Given a set of polynomials Pi(x), i =1:::s of degree at most 2 over GF [p] in n variables, nd a root common to as many as possible of the polynomials Pi(x). We prove that in the case when the polynomials do not contain any squares as monomials, it i ..."
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Cited by 18 (1 self)
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We consider the following NP optimization problem: Given a set of polynomials Pi(x), i =1:::s of degree at most 2 over GF [p] in n variables, nd a root common to as many as possible of the polynomials Pi(x). We prove that in the case when the polynomials do not contain any squares as monomials, it is always possible to approximate this problem within a factor of p2 p;1 in polynomial time for xed p. This follows from the stronger statement that one can, in polynomial time, nd an assignment that satis es at least p;1 p2 of the nontrivial equations. More interestingly, we prove that approximating the maximal number of polynomials with a common root to within a factor of p; is NPhard. This implies that the ratio between the performance of the approximation algorithm and the impossibility result is essentially p p;1 which can be made arbitrarily close to 1 by choosing p large. We also prove that for any constant <1, it is NPhard to approximate the solution of quadratic equations over the rational numbers, or over the reals, within n.
Transparent Proofs and Limits to Approximation
, 1994
"... We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spotchecks. Recent work by a large group of researc ..."
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Cited by 16 (0 self)
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We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spotchecks. Recent work by a large group of researchers has shown that this seemingly paradoxical concept can be formalized and is feasible in a remarkably strong sense; every formal proof in ZF, say, can be rewritten in transparent format (proving the same theorem in a different proof system) without increasing the length of the proof by too much. This result in turn has surprising implications for the intractability of approximate solutions of a wide range of discrete optimization problems, extending the pessimistic predictions of the PNP theory to approximate solvability. We discuss the main results on transparent proofs and their implications to discrete optimization. We give an account of several links between the two subjects as well ...
IBM Almaden
"... Abstract We consider the following NP optimization problem: Given a set of polynomials Pi(x), i = 1: : : s of degree at most 2 over GF [p] in n variables, find a root common to as many as possible of the polynomials Pi(x). We prove that in the case when the polynomials do not contain any squares as ..."
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Abstract We consider the following NP optimization problem: Given a set of polynomials Pi(x), i = 1: : : s of degree at most 2 over GF [p] in n variables, find a root common to as many as possible of the polynomials Pi(x). We prove that in the case when the polynomials do not contain any squares as monomials, it is always possible to approximate this problem within a factor of p 2 p\Gamma 1in polynomial time for fixed p. This follows from the stronger statement that one can, in polynomial time, find an assignment that satisfies at least p\Gamma 1p2 of the nontrivial equations. More interestingly, we prove that approximating the maximal number of polynomials with a common root to within a factor of p \Gamma ffl is NPhard. This implies that the ratio between the performance of the approximation algorithm and the impossibility result is essentially pp\Gamma 1 which can be made arbitrarily close to 1 by choosing p large. We also prove that for any constant ffi! 1, it is NPhard to approximate the solution of quadratic equations over the rational numbers, or over the reals, within nffi.