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39
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 721 (46 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.
Some optimal inapproximability results
, 2002
"... We prove optimal, up to an arbitrary ffl? 0, inapproximability results for MaxEkSat for k * 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for ..."
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Cited by 647 (12 self)
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We prove optimal, up to an arbitrary ffl? 0, inapproximability results for MaxEkSat for k * 3, maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for MaxE2Sat, MaxCut, MaxdiCut, and Vertex cover. Warning: Essentially this paper has been published in JACM and is subject to copyright restrictions. In particular it is for personal use only.
Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 429 (62 self)
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We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being kcolorable or having a aeclique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edgequeries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...
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 206 (41 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.
Testing Monotonicity
, 1999
"... We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : f0; 1g 7! f0; 1g at arguments of its choice, the test always accepts a monotone f , and rejects f with high probability if it is fflfar from being monotone (i.e., e ..."
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Cited by 55 (12 self)
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We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : f0; 1g 7! f0; 1g at arguments of its choice, the test always accepts a monotone f , and rejects f with high probability if it is fflfar from being monotone (i.e., every monotone function differs from f on more than an ffl fraction of the domain).
Testing lowdegree polynomials over GF(2)
 RANDOMAPPROX 2003
"... Abstract. We describe an efficient randomized algorithm to test if a given binary function � � ����� � � ���� � is a lowdegree polynomial (that is, a sum of lowdegree monomials). For a given integer � � � and a given real � � �, the algorithm queries � at � � � � � ��� � points. If � is a ..."
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Cited by 47 (9 self)
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Abstract. We describe an efficient randomized algorithm to test if a given binary function � � ����� � � ���� � is a lowdegree polynomial (that is, a sum of lowdegree monomials). For a given integer � � � and a given real � � �, the algorithm queries � at � � � � � ��� � points. If � is a polynomial of degree at most �, the algorithm always accepts, and if the value of � has to be modified on at least an � fraction of all inputs in order to transform it to such a polynomial, then the algorithm rejects with probability at least ���. Our result is essentially tight: Any algorithm for testing degree � polynomials over � � �� � must perform � � � � � �� � queries. 1
Combinatorial Property Testing (a survey)
 In: Randomization Methods in Algorithm Design
, 1998
"... We consider the question of determining whether a given object has a predetermined property or is "far" from any object having the property. Specifically, objects are modeled by functions, and distance between functions is measured as the fraction of the domain on which the functions differ. We cons ..."
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Cited by 45 (2 self)
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We consider the question of determining whether a given object has a predetermined property or is "far" from any object having the property. Specifically, objects are modeled by functions, and distance between functions is measured as the fraction of the domain on which the functions differ. We consider (randomized) algorithms which may query the function at arguments of their choice, and seek algorithms which query the function at relatively few places. We focus on combinatorial properties, and specifically on graph properties. The two standard representations of graphs  by adjacency matrices and by incidence lists  yield two different models for testing graph properties. In the first model, most appropriate for dense graphs, distance between Nvertex graphs is measured as the fraction of edges on which the graphs disagree over N 2 . In the second model, most appropriate for boundeddegree graphs, distance between Nvertex ddegree graphs is measured as the fraction of edges on ...
Improved Testing Algorithms for Monotonicity
, 1999
"... We present improved algorithms for testing monotonicity of functions. Namely, given theability to query an unknown function f: \Sigma n 7! \Xi, where \Sigma and \Xi are finite ordered sets,the test always accepts a monotone f, and rejects f with high probability if it is fflfar frombeing monotone ( ..."
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Cited by 35 (9 self)
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We present improved algorithms for testing monotonicity of functions. Namely, given theability to query an unknown function f: \Sigma n 7! \Xi, where \Sigma and \Xi are finite ordered sets,the test always accepts a monotone f, and rejects f with high probability if it is fflfar frombeing monotone (i.e., every monotone function differs from f on more than an ffl fraction of thedomain). For any ffl> 0, the query and time complexities of the test are O((n/ffl)*log \Sigma *log \Xi ).The previous best known bound was ~
On the Robustness of Functional Equations
 SIAM Journal on Computing
, 1994
"... In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for ..."
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Cited by 22 (2 self)
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In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for robustness. We then study a general class of functional equations, which are of the form 8x; y F [f(x \Gamma y); f(x + y); f(x); f(y)] = 0, where F is an algebraic function. We give conditions on such functional equations that imply robustness. Our results have applications to the area of selftesting/correcting programs. We show that selftesters and selfcorrectors can be found for many functions satisfying robust functional equations, including algebraic functions of trigonometric functions such as tan x; 1 1+cotx ; Ax 1\GammaAx ; cosh x. 1 Introduction The mathematical field of functional equations is concerned with the following prototypical problem: Given a set of properties (fun...
Testing problems with sublearning sample complexity
 In Proceedings of the Eleventh Annual ACM Conference on Computational Learning Theory
, 1998
"... We study the problem of determining, for a class of functions ¡ , whether an unknown target function ¢ is contained in ¡ or is “far ” from any function in ¡. Thus, in contrast to problems of learning, where we must construct a good approximation to ¢ in ¡ on the basis of sample data, in problems of ..."
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Cited by 17 (11 self)
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We study the problem of determining, for a class of functions ¡ , whether an unknown target function ¢ is contained in ¡ or is “far ” from any function in ¡. Thus, in contrast to problems of learning, where we must construct a good approximation to ¢ in ¡ on the basis of sample data, in problems of testing we are only required to determine the existence of a good approximation. Our main results demonstrate that, over the domain £ ¤ ¥ ¦ § ¨ for constant © , the number of examples required for testing grows only as � � � ��� � � � (where � is any small constant), for both decision trees of size � and a special class of neural networks with � hidden units. This is in contrast to the � � � � examples required for learning these same classes. Our tests are based on combinatorial constructions demonstrating that these classes can be approximated by small classes of coarse partitions of space, and rely on repeated application of the wellknown Birthday Paradox. � Supported by an ONR Science Scholar Fellowship at the Bunting Institute. 1