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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...
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 187 (49 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.
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 157 (25 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.
Quick Approximation to Matrices and Applications
, 1998
"... We give algorithms to find the following simply described approximation to a given matrix. Given an m \Theta n matrix A with entries between say1 and 1, and an error parameter ffl between 0 and 1, we find a matrix D (implicitly) which is the sum of O(1=ffl 2 ) simple rank 1 matrices so that the ..."
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Cited by 151 (7 self)
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We give algorithms to find the following simply described approximation to a given matrix. Given an m \Theta n matrix A with entries between say1 and 1, and an error parameter ffl between 0 and 1, we find a matrix D (implicitly) which is the sum of O(1=ffl 2 ) simple rank 1 matrices so that the sum of entries of any submatrix (among the 2 m+n ) of (A \Gamma D) is at most fflmn in absolute value. Our algorithm takes time dependent only on ffl and the allowed probability of failure (not on m;n). We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph Theory and the constructive version of Alon, Duke, Leffman, Rödl and Yuster. The second one is from the papers of Arora, Karger and Karpinski, Fernandez de la Vega and most directly Goldwasser, Goldreich and Ron who develop approximation algorithms for a set of graph problems, typical of which is the maximum cut problem. ?From our matrix approximation, the...
Approximating the cutnorm via Grothendieck’s inequality
 Proc. of the 36 th ACM STOC
, 2004
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Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs
 Bioinformatics
, 2004
"... Biological and engineered networks have recently been shown to display network motifs: a small set of characteristic patterns which occur much more frequently than in randomized networks with the same degree sequence. Network motifs were demonstrated to play key information processing roles in biolo ..."
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Cited by 121 (0 self)
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Biological and engineered networks have recently been shown to display network motifs: a small set of characteristic patterns which occur much more frequently than in randomized networks with the same degree sequence. Network motifs were demonstrated to play key information processing roles in biological regulation networks. Existing algorithms for detecting network motifs act by exhaustively enumerating all subgraphs with a given number of nodes in the network. The runtime of such full enumeration algorithms increases strongly with network size. Here we present a novel algorithm that allows estimation of subgraph concentrations and detection of network motifs at a run time that is asymptotically independent of the network size. This algorithm is based on random sampling of subgraphs. Network motifs are detected with a surprisingly small number of samples in a wide variety of networks. Our method can be applied to estimate the concentrations of larger subgraphs in larger networks than was previously possible with full enumeration algorithms. We present results for highorder motifs in several biological networks and discuss their possible functions. Availability: A software tool for estimating subgraph concentrations and detecting network motifs (mfinder 2.0) and further information is available at:
The algorithmic aspects of the Regularity Lemma
 J. Algorithms
, 1994
"... The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that ..."
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Cited by 113 (30 self)
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The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that deciding if a given partition of an input graph satisfies the properties guaranteed by the lemma is coNPcomplete. However, we also prove that despite this difficulty the lemma can be made constructive; we show how to obtain, for any input graph, a partition with the properties guaranteed by the lemma, efficiently. The desired partition, for an nvertex graph, can be found in time O(M(n)), where M(n) = O(n 2.376) is the time needed to multiply two n by n matrices with 0, 1entries over the integers. The algorithm can be parallelized and implemented in NC 1. Besides the curious phenomenon of exhibiting a natural problem in which the search for a solution is easy whereas the decision if a given instance is a solution is difficult (if P and NP differ), our constructive version of the Regularity Lemma supplies efficient sequential and parallel algorithms for many problems, some of which are naturally motivated by the study of various graph embedding and graph coloring problems.
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 111 (19 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 85 (15 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.
A New Rounding Procedure for the Assignment Problem with Applications to Dense Graph Arrangement Problems
, 2001
"... We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the wellkn ..."
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Cited by 78 (3 self)
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We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the wellknown LP rounding procedure of Raghavan and Thompson, which is usually used to round fractional solutions of linear programs.