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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 827 (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.
The History and Status of the P versus NP Question
, 1992
"... this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the re ..."
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Cited by 59 (1 self)
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this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies
The Role of Relativization in Complexity Theory
 Bulletin of the European Association for Theoretical Computer Science
, 1994
"... Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and progr ..."
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Cited by 43 (10 self)
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Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and program checking results and try to understand why they do not relativize. We give some new results that may help us to understand these questions better.
TimeSpace Tradeoffs for Satisfiability
 Journal of Computer and System Sciences
, 1997
"... We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved ..."
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Cited by 37 (1 self)
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We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for logspace uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomialtime hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...
Circuit Minimization Problem
 In ACM Symposium on Theory of Computing (STOC
, 1999
"... We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surpris ..."
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Cited by 36 (4 self)
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We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s. We argue why this problem is unlikely to be in P (or even in P=poly) by giving a number of surprising consequences of such an assumption. We also argue that proving this problem to be NPcomplete (if it is indeed true) would imply proving strong circuit lower bounds for the class E, which appears beyond the currently known techniques. Keywords: hard Boolean functions, derandomization, natural properties, NPcompleteness. 1 Introduction An nvariable Boolean function f n : f0; 1g n ! f0; 1g can be given by either its truth table of size 2 n , or a Boolean circuit whose size may be significantly smaller than 2 n . It is well known that most Boolean functions on n variables have circuit complexity at least 2 n =n [Sha49], but so far no family of sufficiently hard functions has ...
The Status of the P versus NP Problem
"... When Moshe Vardi asked me to write this piece for CACM, my first reaction was the article could be written in two words Still open. ..."
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Cited by 24 (0 self)
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When Moshe Vardi asked me to write this piece for CACM, my first reaction was the article could be written in two words Still open.
A Short History of Computational Complexity
 IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2002
"... this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quit ..."
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Cited by 18 (1 self)
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this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld
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 ...
A Taxonomy of Proof Systems
 BASIC RESEARCH IN COMPUTER SCIENCE, CENTER OF THE DANISH NATIONAL RESEARCH FOUNDATION
, 1997
"... Several alternative formulations of the concept of an efficient proof system are nowadays coexisting in our field. These systems include the classical formulation of NP , interactive proof systems (giving rise to the class IP), computationallysound proof systems, and probabilistically checkable pro ..."
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Cited by 16 (2 self)
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Several alternative formulations of the concept of an efficient proof system are nowadays coexisting in our field. These systems include the classical formulation of NP , interactive proof systems (giving rise to the class IP), computationallysound proof systems, and probabilistically checkable proofs (PCP), which are closely related to multiprover interactive proofs (MIP). Although these notions are sometimes introduced using the same generic phrases, they are actually very different in motivation, applications and expressive power. The main objective of this essay is to try to clarify these differences.