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42
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 718 (45 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. 1
Designing Programs That Check Their Work
, 1989
"... A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It d ..."
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Cited by 307 (17 self)
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A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable.
Software Reliability via RunTime ResultChecking
 JOURNAL OF THE ACM
, 1994
"... We review the field of resultchecking, discussing simple checkers and selfcorrectors. We argue that such checkers could profitably be incorporated in software as an aid to efficient debugging and reliable functionality. We consider how to modify traditional checking methodologies to make them more ..."
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Cited by 101 (2 self)
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We review the field of resultchecking, discussing simple checkers and selfcorrectors. We argue that such checkers could profitably be incorporated in software as an aid to efficient debugging and reliable functionality. We consider how to modify traditional checking methodologies to make them more appropriate for use in realtime, realnumber computer systems. In particular, we suggest that checkers should be allowed to use stored randomness: i.e., that they should be allowed to generate, preprocess, and store random bits prior to runtime, and then to use this information repeatedly in a series of runtime checks. In a case study of checking a general realnumber linear transformation (for example, a Fourier Transform), we present a simple checker which uses stored randomness, and a selfcorrector which is particularly efficient if stored randomness is allowed.
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 ac ..."
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Cited by 68 (9 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 questi...
2006, Quantum verification of matrix products
 Proceedings of the 17th ACMSIAM Symposium on Discrete Algorithms
"... We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worstcase time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in ..."
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Cited by 34 (0 self)
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We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worstcase time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in time O(n 7/4). We also present a quantum matrix multiplication algorithm that is efficient when the result has few nonzero entries. 1
Lower Bounds for Oneway Probabilistic Communication Complexity
, 1992
"... this paper can be generalized to the optimal model? 8 Acknowledgment I wish to thank L. Hemachandra for his invitation to me to spend the spring semester at the University of Rochester and for his permanent attention to my research and helpfulness in all my problems and J. Seiferas for extensive c ..."
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Cited by 30 (2 self)
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this paper can be generalized to the optimal model? 8 Acknowledgment I wish to thank L. Hemachandra for his invitation to me to spend the spring semester at the University of Rochester and for his permanent attention to my research and helpfulness in all my problems and J. Seiferas for extensive comments on an earlier draft of this paper. The results of section 4.1 of the paper are the realization of J. Seiferas's advice to investigate the probabilistic complexity properties of almost all functions in comparison with Yao's [Y1] results. I wish also to thank P. Dietz for his comments, which helped to simplify the proof of lemma 4.1
Subquadratic ZeroKnowledge
, 1995
"... We improve on the communication complexity of zeroknowledge proof systems. Let C be a boolean circuit of size n. Previous zeroknowledge proof systems for the satisfiability of C require the use of \Omega\Gamma kn) bit commitments in order to achieve a probability of undetected cheating below 2 \G ..."
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Cited by 13 (3 self)
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We improve on the communication complexity of zeroknowledge proof systems. Let C be a boolean circuit of size n. Previous zeroknowledge proof systems for the satisfiability of C require the use of \Omega\Gamma kn) bit commitments in order to achieve a probability of undetected cheating below 2 \Gammak . In the case k = n, the communication complexity of these protocols is therefore\Omega\Gamma n 2 ) bit commitments. In this paper, we present a zeroknowledge proof system for achieving the same goal with only O(n 1+"n + k p n 1+"n ) bit commitments, where " n goes to zero as n goes to infinity. In the case k = n, this is O(n p n 1+"n ). Moreover, only O(k) commitments need ever be opened, which is interesting if it is substantially less expensive to commit to a bit than to open a commitment. A preliminary version of this paper appeared in the Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, October 1991. y Supported in part by NSA Gr...
Testing Multivariate Linear Functions: Overcoming the Generator Bottleneck
 Proc. 27th STOC
, 1994
"... The problem of testing program correctness has received considerable attention in computer science. One approach to this problem is the notion of selftesting programs [BLR90]. Selftesting usually becomes more costly in the case of testing multivariate functions. In this paper we present efficien ..."
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Cited by 10 (1 self)
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The problem of testing program correctness has received considerable attention in computer science. One approach to this problem is the notion of selftesting programs [BLR90]. Selftesting usually becomes more costly in the case of testing multivariate functions. In this paper we present efficient methods for selftesting multivariate linear functions. We then apply these methods to several multivariate linear problems to construct efficient selftesters. Cornell University. email: ergun@cs.cornell.edu. This work is supported by ONR Young Investigator Award N000149310590 1 1 Introduction Selftesting/correcting programs, which were introduced in [BLR90], are a powerful tool for attacking the problem of program correctness. Various problems have been shown to have selftesters and selfcorrectors[BLR90][BF90][Lip91][CL90][GLRSW91][RS92][RS93]. In this paper we investigate the problem of selftesting multivariate linear functions, i.e., given a multivariate linear function f a...
ikusts. Probabilities to accept languages by quantum finite automata
 In Proceedings of the 5th Annual International Conference on Computing and Combinatorics (COCOON'99), Lecture Notes in Computer Science
, 1999
"... We construct a hierarchy of regular languages such that the current language in the hierarchy can be accepted by 1way quantum finite automata with a probability smaller than the corresponding probability for the preceding language in the hierarchy. These probabilities converge to 1 2. ..."
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Cited by 9 (0 self)
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We construct a hierarchy of regular languages such that the current language in the hierarchy can be accepted by 1way quantum finite automata with a probability smaller than the corresponding probability for the preceding language in the hierarchy. These probabilities converge to 1 2.
A Probabilistic Algorithm for Verifying Matrix Products Using O(n²) Time and log_{2}n+O(1) Random Bits
, 1991
"... A onesided error probabilistic algorithm is given that determines, for n \Theta n input matrices A, B, and C, whether AB 6= C, using O(n 2 ) multiplications and additions and dlog 2 ne + 1 random bits. We further show how to reduce the error probability to ffl with only an additional dlog 2 ( 1 ..."
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Cited by 9 (0 self)
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A onesided error probabilistic algorithm is given that determines, for n \Theta n input matrices A, B, and C, whether AB 6= C, using O(n 2 ) multiplications and additions and dlog 2 ne + 1 random bits. We further show how to reduce the error probability to ffl with only an additional dlog 2 ( 1 ffl )e random bits. This material is based upon work supported under a National Science Foundation Graduate Fellowship. y This material is based upon work supported by the National Science Foundation under grants CCR8858799 and CCR8907960. 1. Introduction Given two n \Theta n matrices A and B, computing their product is a classic problem. We consider a related decision problem: given three n \Theta n matrices A, B, and C, how difficult is it to verify whether AB = C? Freivalds [2] gave a probabilistic algorithm to verify matrix products using O(n 2 ) multiplications and additions, and n bits of randomness. The algorithm accepts the set fhA; B;Ci j AB 6= Cg with onesided e...