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98
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 91 (16 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
A combinatorial characterization of the testable graph properties: it’s all about regularity
 Proc. of STOC 2006
, 2006
"... A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi’s regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédipartition is testable with a constant ..."
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Cited by 69 (14 self)
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A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi’s regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédipartition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédipartitions. This means that in some sense, testing for Szemerédipartitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of propertytesting, which was first raised in the 1996 paper of Goldreich, Goldwasser and Ron [24] that initiated the study of graph propertytesting. This characterization also gives an intuitive explanation as to what makes a graph property testable.
Some 3CNF properties are hard to test
 In Proc. 35th ACM Symp. on Theory of Computing
, 2003
"... Abstract. For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of nbit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that th ..."
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Cited by 58 (11 self)
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Abstract. For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of nbit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that there are 3CNF formulae (with O(n) clauses) such that testing for the associated property requires Ω(n) queries, even with adaptive tests. This contrasts with 2CNF formulae, whose associated properties are always testable with O ( √ n) queries [E. Fischer et al., Monotonicity testing over general poset domains, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 474–483]. Notice that for every negative instance (i.e., an assignment that does not satisfy ϕ) there are three bit queries that witness this fact. Nevertheless, finding such a short witness requires reading a constant fraction of the input, even when the input is very far from satisfying the formula that is associated with the property. A property is linear if its elements form a linear space. We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include the following observations which are of independent interest: 1. In the context of testing for linear properties, adaptive twosided error tests have no more power than nonadaptive onesided error tests. Moreover, without loss of generality, any test for a linear property is a linear test. A linear test verifies that a portion of the input satisfies a set of linear constraints, which define the property, and rejects if and only if it finds a falsified constraint. A linear test is by definition nonadaptive and, when applied to linear properties, has a onesided error. 2. Random low density parity check codes (which are known to have linear distance and constant rate) are not locally testable. In fact, testing such a code of length n requires Ω(n) queries.
Testing Subgraphs in Directed Graphs
 Proc. of the 35 th Annual Symp. on Theory of Computing (STOC
, 2003
"... Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least #n edges have to be deleted from it to make it Hfree. We show that in this case G contains at least f(#, H)n copies of H. This is proved by establishing a directed version of Sz ..."
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Cited by 51 (17 self)
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Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least #n edges have to be deleted from it to make it Hfree. We show that in this case G contains at least f(#, H)n copies of H. This is proved by establishing a directed version of Szemeredi's regularity lemma, and implies that for every H there is a onesided error property tester whose query complexity is bounded by a function of # only for testing the property PH of being Hfree.
Monotonicity testing over general poset domains (Extended Abstract)
 STOC'02
, 2002
"... The field of property testing studies algorithms that distinguish, using a small number of queries, between inputs which satisfy a given property, and those that are ‘far’ from satisfying the property. Testing properties that are defined in terms of monotonicity has been extensively investigated, pr ..."
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Cited by 48 (22 self)
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The field of property testing studies algorithms that distinguish, using a small number of queries, between inputs which satisfy a given property, and those that are ‘far’ from satisfying the property. Testing properties that are defined in terms of monotonicity has been extensively investigated, primarily in the context of the monotonicity of a sequence of integers, or the monotonicity of a function over the £dimensional hypercube ¤¥¦§§ § ¦¨©�. These works resulted in monotonicity testers whose query complexity is at most polylogarithmic in the size of the domain. We show that in its most general setting, testing that Boolean functions are close to monotone is equivalent, with respect to the number of required queries, to several other testing problems in logic and graph theory. These problems include: testing that a Boolean assignment of variables is close to an assignment that satisfies a specific �CNF formula, testing that a set of vertices is close to one that is a vertex cover of a specific graph, and testing that a set of vertices is close to a clique. We then investigate the query complexity of monotonicity testing of both Boolean and integer functions over general partial orders. We give algorithms and lower bounds for the general problem, as well as for some interesting special cases. In proving a general lower bound, we construct graphs with combinatorial properties that may be of independent interest.
Testing Juntas
, 2002
"... We show that a Boolean function over n Boolean variables can be tested for the property of depending on only k of them, using a number of queries that depends only on k and the approximation parameter . We present two tests, both nonadaptive, that require a number of queries that is polynomial k an ..."
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Cited by 46 (8 self)
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We show that a Boolean function over n Boolean variables can be tested for the property of depending on only k of them, using a number of queries that depends only on k and the approximation parameter . We present two tests, both nonadaptive, that require a number of queries that is polynomial k and linear in . The first test is stronger in that it has a 1sided error, while the second test has a more compact analysis. We also present an adaptive version and a 2sided error version of the first test, that have a somewhat better query complexity than the other algorithms...
Every monotone graph property is testable
 Proc. of STOC 2005
, 2005
"... A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most wellstudied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper i ..."
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Cited by 43 (9 self)
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A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most wellstudied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with onesided error, and with query complexity depending only on ɛ. This result unifies several previous results in the area of property testing, and also implies the testability of wellstudied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerédi’s Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with onesided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graphtheory, is that for any monotone graph property P, any graph that is ɛfar from satisfying P, contains a subgraph of size depending on ɛ only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is ɛfar from satisfying a (possibly infinite) set of monotone graph properties P, then it is at least δP(ɛ)far from satisfying one of the properties.
Graph limits and parameter testing
"... We define a distance of two graphs that reflects the closeness of both local and global properties. We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form ..."
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Cited by 37 (1 self)
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We define a distance of two graphs that reflects the closeness of both local and global properties. We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing. As examples, we provide short proofs of the testability of MaxCut and the recent result of Alon and Shapira about the testability of hereditary graph properties.
Testing Basic Boolean Formulae
 SIAM J. Disc. Math
, 2002
"... We consider the problem of determining whether a given function f : f0; 1g belongs to a certain class of Boolean functions F or whether it is far from the class. More precisely, given query access to the function f and given a distance parameter , we would like to decide whether f 2 F or whethe ..."
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Cited by 36 (6 self)
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We consider the problem of determining whether a given function f : f0; 1g belongs to a certain class of Boolean functions F or whether it is far from the class. More precisely, given query access to the function f and given a distance parameter , we would like to decide whether f 2 F or whether it diers from every g 2 F on more than an fraction of the domain elements. The classes of functions we consider are singleton (\dictatorship") functions, monomials, and monotone DNF functions with a bounded number of terms. In all cases we provide algorithms whose query complexity is independent of n (the number of function variables), and linear in 1=.
Testing versus estimation of graph properties
 Proc. of STOC 2005
, 2005
"... In the course of the proof we develop a framework for extending Szemer'edi's Regularity Lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. In particular ..."
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Cited by 29 (7 self)
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In the course of the proof we develop a framework for extending Szemer'edi's Regularity Lemma, both as a prerequisite for formulating what kind of information about the input graph will provide us with the correct estimation, and as the means for efficiently gathering this information. In particular, we construct a probabilistic algorithm that finds the parameters of a regular partition of an input graph using a constant number of queries, and an algorithm to find a regular partition of a graph using a TC0 circuit. This, in some ways, strengthens the results of [1].