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349
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 754 (38 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.
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 461 (67 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...
Efficient Testing of Large Graphs
 Combinatorica
"... Let P be a property of graphs. An test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it h ..."
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Cited by 168 (47 self)
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Let P be a property of graphs. An test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than n 2 edges to make it satisfy P . The property P is called testable, if for every there exists an test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain graph properties admit an test. In this paper we make a first step towards a logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type "89" are always testable, while we show that some properties containing this alternation are not. Our results are proven using a combinatorial lemma, a special case of which, that may be of independent interest, is the following. A graph H is called unavoidable in G if all graphs that differ from G in no more than jGj 2 places contain an induced copy of H . A graph H is called abundant in G if G contains at least jGj jHj induced copies of H. If H is unavoidable in G then it is also ( ; jHj)abundant.
Fast batch verification for modular exponentiation and digital signatures
, 1998
"... Abstract Many tasks in cryptography (e.g., digital signature verification) call for verification of a basicoperation like modular exponentiation in some group: given ( g, x, y) check that gx = y. Thisis typically done by recomputing gx and checking we get y. We would like to do it differently,and f ..."
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Cited by 144 (2 self)
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Abstract Many tasks in cryptography (e.g., digital signature verification) call for verification of a basicoperation like modular exponentiation in some group: given ( g, x, y) check that gx = y. Thisis typically done by recomputing gx and checking we get y. We would like to do it differently,and faster. The approach we use is batching. Focusing first on the basic modular exponentiation operation, we provide some probabilistic batch verifiers, or tests, that verify a sequence of modular exponentiations significantly faster than the naive recomputation method. This yields speedupsfor several verification tasks that involve modular exponentiations.
Improved lowdegree testing and its applications
 IN 29TH STOC
, 1997
"... NP = PCP(log n, 1) and related results crucially depend upon the close connection betsveen the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [29]. The stro ..."
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Cited by 144 (18 self)
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NP = PCP(log n, 1) and related results crucially depend upon the close connection betsveen the probability with which a function passes a low degree test and the distance of this function to the nearest degree d polynomial. In this paper we study a test proposed by Rubinfeld and Sudan [29]. The strongest previously known connection for this test states that a function passes the test with probability 6 for some d> 7/8 iff the function has agreement N 6 with a polynomial of degree d. We presenta new, and surprisingly strong,analysiswhich shows thatthepreceding statementis truefor 6<<0.5. The analysis uses a version of Hilbe?l irreducibility, a tool used in the factoring of multivariate polynomials. As a consequence we obtain an alternate construction for the following proof system: A constant prover lround proof system for NP languages in which the verifier uses O(log n) random bits, receives answers of size O(log n) bits, and has an error probability of at most 2 – 10g*‘’. Such a proof system, which implies the NPhardness of approximating Set Cover to within fl(log n) factors, has already been obtained by Raz and Safra [28]. Our result was completed after we heard of their claim. A second consequence of our analysis is a self testerlcorrector for any buggy program that (supposedly) computes a polynomial over a finite field. If the program is correct only on 6 fraction of inputs where 15<<0.5, then the tester/corrector determines J and generates 0(~) randomized programs, such that one of the programs is correct on every input, with high probability.
The art of uninformed decisions: A primer to property testing
 Science
, 2001
"... Property testing is a new field in computational theory, that deals with the information that can be deduced from the input where the number of allowable queries (reads from the input) is significally smaller than its size. ..."
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Cited by 143 (23 self)
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Property testing is a new field in computational theory, that deals with the information that can be deduced from the input where the number of allowable queries (reads from the input) is significally smaller than its size.
Property Testing in Bounded Degree Graphs
 Algorithmica
, 1997
"... We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength in ..."
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Cited by 129 (39 self)
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We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of boundeddegree graphs. In particular, we present randomized algorithms for testing whether an unknown boundeddegree graph is connected, kconnected (for k ? 1), planar, etc. Our algorithms work in time polynomial in 1=ffl, always accept the graph when it has the tested property, and reject with high probability if the graph is fflaway from having the property. For example, the 2Connectivity algorithm rejects (w.h.p.) any Nvertex ddegree graph for which more ...
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 114 (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.
Learning polynomials with queries: The highly noisy case
, 1995
"... Given a function f mapping nvariate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of agfieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of allnvariate degreedpolynomials which agree withf ..."
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Cited by 94 (18 self)
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Given a function f mapping nvariate inputs from a finite Kearns et. al. [21] (see also [27, 28, 22]). In the setting of agfieldFintoF, we consider the task of reconstructing a list nostic learning, the learner is to make no assumptions regarding of allnvariate degreedpolynomials which agree withfon a the natural phenomena underlying the input/output relationship tiny but nonnegligible fraction, , of the input space. We give a of the function, and the goal of the learner is to come up with a randomized algorithm for solving this task which accessesfas a simple explanation which best fits the examples. Therefore the black box and runs in time polynomial in1;nand exponential in best explanation may account for only part of the phenomena. d, provided is(pd=jFj). For the special case whend=1, In some situations, when the phenomena appears very irregular, we solve this problem for jFj>0. In this case the providing an explanation which fits only part of it is better than nothing. Interestingly, Kearns et. al. did not consider the use of running time of our algorithm is bounded by a polynomial queries (but rather examples drawn from an arbitrary distribuand exponential ind. Our algorithm generalizes a previously tion) as they were skeptical that queries could be of any help. known algorithm, due to Goldreich and Levin, that solves this We show that queries do seem to help (see below). task for the case whenF=GF(2)(andd=1).
A characterization of the (natural) graph properties testable with onesided error
 Proc. of FOCS 2005
, 2005
"... The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of propertytesting. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decis ..."
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Cited by 94 (17 self)
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The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of propertytesting. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property P has an oblivious onesided error tester, if and only if P is (almost) hereditary. We stress that any ”natural ” property that can be tested (either with onesided or with twosided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the ”natural” graph properties, which are testable with onesided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with onesided error. This general result contains as a special case all the previous results about testing graph properties with onesided error. These include the results of [20] and [5] about testing kcolorability, the characterization of [21] of the graphpartitioning problems that are testable with onesided error, the induced vertex colorability properties of [3], the induced edge colorability properties of [14], a transformation from twosided to onesided error testing [21], as well as a recent result about testing monotone graph properties [10]. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with onesided error. Some of these properties are the well known graph properties of being Perfect, Chordal, Interval, Comparability, Permutation and more. None of these properties was previously known to be testable. 1