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22
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 ..."
Abstract

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.
Probabilistic checking of proofs: a new characterization of NP
 JOURNAL OF THE ACM
, 1998
"... 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 ..."
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Cited by 414 (26 self)
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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.
Free Bits, PCPs and NonApproximability  Towards Tight Results
, 1996
"... This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems. ..."
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Cited by 212 (39 self)
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This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems.
The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations
, 1993
"... We prove the following about the Nearest Lattice Vector Problem (in any `p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NPhard. 2. If for some ..."
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Cited by 170 (7 self)
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We prove the following about the Nearest Lattice Vector Problem (in any `p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NPhard. 2. If for some ffl ? 0 there exists a polynomialtime algorithm that approximates the optimum within a factor of 2 log 0:5\Gammaffl n , then every NP language can be decided in quasipolynomial deterministic time, i.e., NP ` DTIME(n poly(log n) ). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the `1 norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2 log 1\Gammaffl n or n ffl . Improving the factor 2 log 0:5\Gammaffl n to p dimension for either of the lattice problems would imply the hardness of the Shortest Vector Problem in `2 norm; an old open problem. Our proofs use reductions from fewpr...
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.
On the complexity of space bounded interactive proofs
, 2005
"... We prove two results on interactive proof systems with 2way probabilistic finite state verifiers. The first is a lower bound on the power of such proof systems, if they are not required to halt with high probability on rejected inputs: we show that they can accept any recursively enumerable languag ..."
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Cited by 78 (5 self)
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We prove two results on interactive proof systems with 2way probabilistic finite state verifiers. The first is a lower bound on the power of such proof systems, if they are not required to halt with high probability on rejected inputs: we show that they can accept any recursively enumerable language. The second is an upper bound on the power of interactive proof systems that halt with high probability on all inputs: any language they accept is in ATIME(22 O(n)). Our results generalize to other space bounds. The proof techniques we develop have other interesting applications. The proof method for the lower bound also shows that the emptiness problem for 1way probabilistic finite state machines is undecidable. In proving the upper bound, we obtain some results of independent interest on the rate of convergence of timevarying Markov chains and of nonMarkov chains, called feedback chains, to their halting states. 1
The complexity of approximating a nonlinear program
 IBM Research Report RC 17831
, 1992
"... We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadra ..."
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Cited by 58 (3 self)
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We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasipolynomial time. Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.
Probabilistically Checkable Debate Systems and Nonapproximability of PSPACEHard Functions
, 1995
"... ..."
Interactive proofs and approximation: reduction from two provers in one round
 In The second Israeli Symposium on the Theory of Computing
, 1993
"... We present hard to approximate problems in the following areas: systems of representatives, network ow, and longest paths in graphs. In each case we show that there exists some>0 such that polynomial time approximation to within a factor of 2log n of the optimal implies that NP has quasi polynomi ..."
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Cited by 18 (1 self)
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We present hard to approximate problems in the following areas: systems of representatives, network ow, and longest paths in graphs. In each case we show that there exists some>0 such that polynomial time approximation to within a factor of 2log n of the optimal implies that NP has quasi polynomial time deterministic simulations. The results are derived by reduction from two prover, one round proof systems, and exemplify the ability ofsuchreductions to yield hardness of approximations results for many di erent kinds of problems.