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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 498 (68 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...
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 78 (15 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).
Property Testing
 Handbook of Randomized Computing, Vol. II
, 2000
"... this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well). ..."
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Cited by 75 (10 self)
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this technical aspect (as in the boundeddegree model the closest graph having the property must have at most dN edges and degree bound d as well).
Gowers uniformity, influence of variables, and PCPs
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We pro ..."
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Cited by 65 (2 self)
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Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We prove the following inverse theorem. Write G = G1 × · · · × Gn as a product of groups. If a bounded balanced function f: G1 × · · · Gn → C is such that U d (f) ≥ ε, then one of the coordinates of f has influence at least ε/2 O(d). Other inverse theorems are known [Gow98, Gow01, GT05, Sam05], and U 3 is especially well understood, but the properties of functions f with large U d (f), d ≥ 4, are not yet well characterized. The dimensiond Gowers inner product 〈{fS} 〉 U d of a collection {fS} S⊆[d] of functions is a related measure of pseudorandomness. The definition is such that if all the functions fS are equal to the same fixed function f, then 〈{fS} 〉 U d = U d (f). We prove that if fS: G1 × · · · × Gn → C is a collection of bounded functions such that 〈{fS} 〉 U d  ≥ ε and at least one of the fS is balanced, then there is a variable that has influence at least ε 2 /2 O(d) for at least four functions in the collection. Finally, we relate the acceptance probability of the “hypergraph longcode test ” proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce the following result: if the Unique Games Conjecture is true, then for every q ≥ 3 there is a PCP characterization of NP where the verifier makes q queries, has almost perfect completeness, and soundness at most 2q/2 q. For infinitely many q, the soundness is (q + 1)/2 q, which might be a tight result. Two applications of this results are that, assuming that the unique games conjecture is true, it is hard to approximate Max kCSP within a factor 2k/2 k ((k + 1)/2 k for infinitely many k), and it is hard to approximate Independent Set in graphs of degree D within a factor (log D) O(1) /D. 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 diffe ..."
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Cited by 53 (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 ...
Combinatorial Approximation Algorithms for the Maximum Directed Cut Problem
 Proc. of 12th SODA
, 2001
"... We describe several combinatorial algorithms for the maximum directed cut problem. Among our results is a simple linear time 9/20approximation algorithm for the problem, and a somewhat slower 1/2approximation algorithm that uses a bipartite matching routine. No better combinatorial approximation a ..."
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Cited by 19 (0 self)
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We describe several combinatorial algorithms for the maximum directed cut problem. Among our results is a simple linear time 9/20approximation algorithm for the problem, and a somewhat slower 1/2approximation algorithm that uses a bipartite matching routine. No better combinatorial approximation algorithms are known even for the easier maximum cut problem for undirected graphs. Our algorithms do not use linear programming, nor semidefinite programming. They are based on the observation that the maximum directed cut problem is equivalent to the problem of finding a maximum independent set in the line graph of the input graph, and that the linear programming relaxation of the problem is equivalent to the problem of finding a maximum fractional independent set of that line graph. The maximum fractional independent set problem can be easily reduced to a bipartite matching problem. As a consequence of this relation, we also get that the maximum directed cut problem for bipartite digraphs can be solved in polynomial time.
Exact and approximate testing/correcting of algebraic functions: A survey
 Electronic Colloq. on Comp. Compl., Univ. of Trier TR2001014
, 2001
"... In the late 80’s Blum, Luby, Rubinfeld, Kannan et al. pioneered the theory of self–testing as an alternative way of dealing with the problem of software reliability. Over the last decade this theory played a crucial role in the construction of probabilistically checkable proofs and the derivation ..."
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Cited by 10 (2 self)
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In the late 80’s Blum, Luby, Rubinfeld, Kannan et al. pioneered the theory of self–testing as an alternative way of dealing with the problem of software reliability. Over the last decade this theory played a crucial role in the construction of probabilistically checkable proofs and the derivation of hardness of approximation results. Applications in areas like computer vision, machine learning, and self–correcting programs were also established. In the self–testing problem one is interested in determining (maybe probabilistically) whether a function to which one has oracle access satisfies a given property. We consider the problem of testing algebraic functions and survey over a decade of research in the area. Special emphasis is given to illustrate the scenario where the problem takes place and to the main techniques used in the analysis of tests. A novel aspect of this work is the separation it advocates between the mathematical and algorithmic issues that arise in the theory of self–testing.
The NonApproximability of NonBoolean Predicates
 Trevisan (Eds.), Proceedings of 5th International Workshop on Randomization and Approximation Techniques in Computer Science
, 2001
"... Constraint satisfaction programs where each constraint depends on a constant number of variables have the following property: The randomized algorithm that guesses an assignment uniformly at random satisfies an expected constant fraction of the constraints. By combining constructions from interactiv ..."
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Cited by 9 (2 self)
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Constraint satisfaction programs where each constraint depends on a constant number of variables have the following property: The randomized algorithm that guesses an assignment uniformly at random satisfies an expected constant fraction of the constraints. By combining constructions from interactive proof systems with harmonic analysis over finite groups, Hstad showed that for several constraint satisfaction programs this naive algorithm is essentially the best possible unless P = NP.