Results 1  10
of
45
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
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 208 (40 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.
Computational Limitations on Learning from Examples
 Journal of the ACM
, 1988
"... Abstract. The computational complexity of learning Boolean concepts from examples is investigated. It is shown for various classes of concept representations that these cannot be learned feasibly in a distributionfree sense unless R = NP. These classes include (a) disjunctions of two monomials, (b) ..."
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Cited by 192 (10 self)
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Abstract. The computational complexity of learning Boolean concepts from examples is investigated. It is shown for various classes of concept representations that these cannot be learned feasibly in a distributionfree sense unless R = NP. These classes include (a) disjunctions of two monomials, (b) Boolean threshold functions, and (c) Boolean formulas in which each variable occurs at most once. Relationships between learning of heuristics and finding approximate solutions to NPhard optimization problems are given. Categories and Subject Descriptors: F. 1.1 [Computation by Abstract Devices]: Models of Computationrelations among models; F. 1.2 [Computation by Abstract Devices]: Modes of Computationprobabilistic computation; F. 1.3 [Computation by Abstract Devices]: Complexity Classesreducibility and completeness; 1.2.6 [Artificial Intelligence]: Learningconcept learning; induction
Zero Knowledge and the Chromatic Number
 Journal of Computer and System Sciences
, 1996
"... We present a new technique, inspired by zeroknowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph. To illustrate this technique we present simple reductions from max3coloring and max3sat, showing that it is hard to approximate the chromatic number wi ..."
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Cited by 178 (8 self)
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We present a new technique, inspired by zeroknowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph. To illustrate this technique we present simple reductions from max3coloring and max3sat, showing that it is hard to approximate the chromatic number within \Omega\Gamma N ffi ), for some ffi ? 0. We then apply our technique in conjunction with the probabilistically checkable proofs of Hastad, and show that it is hard to approximate the chromatic number to within\Omega\Gamma N 1\Gammaffl ) for any ffl ? 0, assuming NP 6` ZPP. Here, ZPP denotes the class of languages decidable by a random expected polynomialtime algorithm that makes no errors. Our result matches (up to low order terms) the known gap for approximating the size of the largest independent set. Previous O(N ffi ) gaps for approximating the chromatic number (such as those by Lund and Yannakakis, and by Furer) did not match the gap for independent set, and do not extend...
Interactive proofs and the hardness of approximating cliques
 Journal of the ACM
, 1996
"... The contribution of this paper is twofold. First, a connection is shown between approximating the size of the largest clique in a graph and multiprover interactive proofs. Second, an efficient multiprover interactive proof for NP languages is constructed, where the verifier uses very few random b ..."
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Cited by 152 (10 self)
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The contribution of this paper is twofold. First, a connection is shown between approximating the size of the largest clique in a graph and multiprover interactive proofs. Second, an efficient multiprover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multiprover interactive proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph. Of independent interest is our proof of correctness for the multilinearity test of functions. 1
Register Allocation via Graph Coloring
, 1992
"... Chaitin and his colleagues at IBM in Yorktown Heights built the first global register allocator based on graph coloring. This thesis describes a series of improvements and extensions to the Yorktown allocator. There are four primary results: Optimistic coloring Chaitin's coloring heuristic pessimis ..."
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Cited by 135 (4 self)
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Chaitin and his colleagues at IBM in Yorktown Heights built the first global register allocator based on graph coloring. This thesis describes a series of improvements and extensions to the Yorktown allocator. There are four primary results: Optimistic coloring Chaitin's coloring heuristic pessimistically assumes any node of high degree will not be colored and must therefore be spilled. By optimistically assuming that nodes of high degree will receive colors, I often achieve lower spill costs and faster code; my results are never worse. Coloring pairs The pessimism of Chaitin's coloring heuristic is emphasized when trying to color register pairs. My heuristic handles pairs as a natural consequence of its optimism. Rematerialization Chaitin et al. introduced the idea of rematerialization to avoid the expense of spilling and reloading certain simple values. By propagating rematerialization information around the SSA graph using a simple variation of Wegman and Zadeck's constant propag...
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 101 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
The minimum consistent DFA problem cannot be approximated within any polynomial
 Journal of the Association for Computing Machinery
, 1993
"... Abstract. The minimum consistent DFA problem is that of finding a DFA with as few states as possible that is consistent with a given sample (a finite collection of words, each labeled as to whether the DFA found should accept or reject). Assuming that P # NP, it is shown that for any constant k, no ..."
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Cited by 82 (4 self)
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Abstract. The minimum consistent DFA problem is that of finding a DFA with as few states as possible that is consistent with a given sample (a finite collection of words, each labeled as to whether the DFA found should accept or reject). Assuming that P # NP, it is shown that for any constant k, no polynomialtime algorithm can be guaranteed to find a consistent DFA with fewer than opt ~ states, where opt is the number of states in the minimum state DFA consistent with the sample. This result holds even if the alphabet is of constant size two, and if the algorithm is allowed to produce an NFA, a regular expression, or a regular grammar that is consistent with the sample. A similar nonapproximability result is presented for the problem of finding small consistent linear grammars. For the case of finding minimum consistent DFAs when the alphabet is not of constant size but instead is allowed to vay with the problem specification, the slightly
On the Hardness of Approximating the Chromatic Number
, 1993
"... We study the hardness of approximating the chromatic number when the input graph is kcolorable for some fixed k 3. Our main result is that it is NPhard to find a 4coloring of a 3chromatic graph. As an immediate corollary we obtain that it is NPhard to color a kchromatic graph with at most ..."
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Cited by 72 (6 self)
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We study the hardness of approximating the chromatic number when the input graph is kcolorable for some fixed k 3. Our main result is that it is NPhard to find a 4coloring of a 3chromatic graph. As an immediate corollary we obtain that it is NPhard to color a kchromatic graph with at most k + 2bk=3c 1 colors. We also give simple proofs of two results of Lund and Yannakakis [20]. The first result shows that it is NPhard to approximate the chromatic number to within n for some fixed > 0. We point