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36
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 128 (20 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.
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
Three Theorems regarding Testing Graph Properties
, 2001
"... Property testing is a relaxation of decision problems in which it is required to distinguish yesinstances (i.e., objects having a predetermined property) from instances that are far from any yesinstance. We presents three theorems regarding testing graph properties in the adjacency matrix represe ..."
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Cited by 73 (10 self)
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Property testing is a relaxation of decision problems in which it is required to distinguish yesinstances (i.e., objects having a predetermined property) from instances that are far from any yesinstance. We presents three theorems regarding testing graph properties in the adjacency matrix representation. More specifically, these theorems relate to the project of characterizing graph properties according to the complexity of testing them (in the adjacency matrix representation). The first theorem is that there exist monotone graph properties in N P for which testing is very hard (i.e., requires to examine a constant fraction of the entries in the matrix). The second theorem is that every graph property that can be tested making a number of queries that is independent of the size of the graph, can be so tested by uniformly selecting a set of vertices and accepting iff the induced subgraph has some fixed graph property (which is not necessarily the same as the one being tested). The third theorem refers to the framework of graph partition problems, and is a characterization of the subclass of properties that can be tested using a onesided error tester making a number of queries that is independent of the size of the graph.
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.
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.
A characterization of easily testable induced subgraphs
 In Proceedings of the Fifteenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2004
"... Let H be a fixed graph on h vertices. We say that a graph G is induced Hfree if it does not contain any induced copy of H. Let G be a graph on n vertices and suppose that at least ɛn 2 edges have to be added to or removed from it in order to make it induced Hfree. It was shown in [5] that in this ..."
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Cited by 24 (11 self)
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Let H be a fixed graph on h vertices. We say that a graph G is induced Hfree if it does not contain any induced copy of H. Let G be a graph on n vertices and suppose that at least ɛn 2 edges have to be added to or removed from it in order to make it induced Hfree. It was shown in [5] that in this case G contains at least f(ɛ, h)n h induced copies of H, where 1/f(ɛ, h) is an extremely fast growing function in 1/ɛ, that is independent of n. As a consequence, it follows that for every H, testing induced Hfreeness with onesided error has query complexity independent of n. A natural question, raised by the first author in [1], is to decide for which graphs H the function 1/f(ɛ, H) can be bounded from above by a polynomial in 1/ɛ. An equivalent question is for which graphs H, can one design a onesided error property tester for testing induced Hfreeness, whose query complexity is polynomial in 1/ɛ. We settle this question almost completely by showing that, quite surprisingly, for any graph other than the paths of lengths 1,2 and 3, the cycle of length 4, and their complements, no such property tester exists. We further show that a similar result also applies to the case of directed graphs, thus answering a question raised by the authors in [9]. We finally show that the same results hold even in the case of twosided error property testers. The proofs combine combinatorial, graph theoretic and probabilistic arguments with results from additive number theory.
Sublinear time algorithms
 SIGACT News
, 2003
"... Abstract Sublinear time algorithms represent a new paradigm in computing, where an algorithmmust give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1 Introduction The goal of algo ..."
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Cited by 22 (2 self)
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Abstract Sublinear time algorithms represent a new paradigm in computing, where an algorithmmust give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. 1 Introduction The goal of algorithmic research is to design efficient algorithms, where efficiency is typicallymeasured as a function of the length of the input. For instance, the elementary school algorithm for multiplying two n digit integers takes roughly n2 steps, while more sophisticated algorithmshave been devised which run in less than n log2 n steps. It is still not known whether a linear time algorithm is achievable for integer multiplication. Obviously any algorithm for this task, as for anyother nontrivial task, would need to take at least linear time in n, since this is what it would take to read the entire input and write the output. Thus, showing the existence of a linear time algorithmfor a problem was traditionally considered to be the gold standard of achievement. Nevertheless, due to the recent tremendous increase in computational power that is inundatingus with a multitude of data, we are now encountering a paradigm shift from traditional computational models. The scale of these data sets, coupled with the typical situation in which there is verylittle time to perform our computations, raises the issue of whether there is time to consider any more than a miniscule fraction of the data in our computations? Analogous to the reasoning thatwe used for multiplication, for most natural problems, an algorithm which runs in sublinear time must necessarily use randomization and must give an answer which is in some sense imprecise.Nevertheless, there are many situations in which a fast approximate solution is more useful than a slower exact solution.
Information theory in property testing and monotonicity testing in higher dimension
 Information and Computation
, 2006
"... Abstract. In general property testing, we are given oracle access to a function f, and we wish to randomly test if the function satisfies a given property P, or it is εfar from having that property. In a more general setting, the domain on which the function is defined is equipped with a probabilit ..."
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Cited by 20 (2 self)
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Abstract. In general property testing, we are given oracle access to a function f, and we wish to randomly test if the function satisfies a given property P, or it is εfar from having that property. In a more general setting, the domain on which the function is defined is equipped with a probability distribution, which assigns different weight to different elements in the distance function. This paper relates the complexity of testing the monotonicity of a function over the ddimensional cube to the Shannon entropy of the underlying distribution. We provide an improved upper bound on the property tester query complexity and we finetune the exponential dependence on the dimension d. 1
Testing hereditary properties of nonexpanding boundeddegree graphs
"... We study graph properties which are testable for bounded degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is believed that almost all, even very simpl ..."
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Cited by 19 (7 self)
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We study graph properties which are testable for bounded degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is believed that almost all, even very simple graph properties require a large complexity to be tested for arbitrary (bounded degree) graphs. Therefore in this paper we focus our attention on testing graph properties for special classes of graphs. We call a graph family nonexpanding if every graph in this family has a weak expansion (its expansion is O(1 / log 2 n), where n is the graph size). A graph family is hereditary if it is closed under vertex removal. Similarly, a graph property is hereditary if it is closed under vertex removal. We call a graph property Π to be testable for a graph family F if for every graph G ∈ F, in time independent of the size of G we can distinguish between the case when G satisfies property Π and when it is far from every graph satisfying property Π. In this paper we prove that in the bounded degree graph model, any hereditary property is testable if the input graph belongs to a hereditary and nonexpanding family of graphs. As an application, our result implies that, for example, any hereditary property (e.g., kcolorability,
Estimating the distance to a monotone function
 Proc. 8th RANDOM
, 2004
"... In standard property testing, the task is to distinguish between objects that have a property P and those that are εfar from P, for some ε> 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy P. This implies that pr ..."
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Cited by 19 (4 self)
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In standard property testing, the task is to distinguish between objects that have a property P and those that are εfar from P, for some ε> 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy P. This implies that property testing in and of itself cannot be expected to yield any information whatsoever about the distance from the object to the property. We address this problem in this paper, restricting our attention to monotonicity testing. A function f: {1,..., n} ↦ → R is at distance εf from being monotone if it can (and must) be modified at εfn places to become monotone. For any fixed δ> 0, we compute, with probability at least 2/3, an interval [(1/2−δ)ε, ε] that encloses εf. The running time of our algorithm is O(ε −1 f log log ε −1 f log n), which is optimal within a factor of log log ε −1 f and represents a substantial improvement over previous work. We give a second algorithm with an expected running time of O(ε −1 f log n log log log n). Finally, we extend our results to multivariate functions. 1