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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 421 (57 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...
Szemerédi's Regularity Lemma and Its Applications in Graph Theory
, 1996
"... Szemer'edi's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by randomlooking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. Recently q ..."
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Cited by 209 (3 self)
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Szemer'edi's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by randomlooking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. Recently quite a few new results were obtained by using the Regularity Lemma, and also some new variants and generalizations appeared. In this survey we describe some typical applications and some generalizations. Contents Preface 1. Introduction 2. How to apply the Regularity Lemma 3. Early applications 4. Building large subgraphs 5. Embedding trees 6. Bounded degree spanning subgraphs 7. Weakening the Regularity Lemma 8. Strengthening the Regularity Lemma 9. Algorithmic questions 10. Regularity and randomness Preface Szemer'edi's Regularity Lemma [121] is one of the most powerful tools of (extremal) graph theory. It was invented as an auxiliary lemma in the proof of the famous conjectu...
The Hardness Of Approximation: Gap Location
 Computational Complexity
, 1994
"... . We refine the complexity analysis of approximation problems by relating it to a new parameter called gap location. Many of the results obtained so far for approximations yield satisfactory analysis with respect to this refined parameter, but some known results (e.g., max kcolorability, max 3dim ..."
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Cited by 82 (0 self)
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. We refine the complexity analysis of approximation problems by relating it to a new parameter called gap location. Many of the results obtained so far for approximations yield satisfactory analysis with respect to this refined parameter, but some known results (e.g., max kcolorability, max 3dimensional matching and max notallequal 3sat) fall short of doing so. As a second contribution, our work fills the gap in these cases by presenting new reductions. Next, we present definitions and hardness results of new approximation versions of some NPcomplete optimization problems. The problems we treat are vertex cover (for which we define a different optimization problem from the one treated in Papadimitriou & Yannakakis 1991), kedge coloring, and set splitting.
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 76 (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).
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 70 (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:
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.
Hfactors in dense graphs
 Graphs and Combinatorics 8
, 1996
"... The following asymptotic result is proved. For every fixed graph H with h vertices, any graph G with n vertices and with minimum degree d ≥ χ(H)−1 χ(H) n contains (1 − o(1))n/h vertex disjoint copies of H. ..."
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Cited by 57 (5 self)
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The following asymptotic result is proved. For every fixed graph H with h vertices, any graph G with n vertices and with minimum degree d ≥ χ(H)−1 χ(H) n contains (1 − o(1))n/h vertex disjoint copies of H.
Szemerédi’s regularity lemma for sparse graphs
 Foundations of Computational Mathematics
, 1997
"... A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This farreaching result has proved to play a central r^ole in many areas of combinatorics, both `pure ' and `algorithmic. ' The quest for ..."
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Cited by 56 (20 self)
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A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This farreaching result has proved to play a central r^ole in many areas of combinatorics, both `pure ' and `algorithmic. ' The quest for an equally powerful variant of this lemma for sparse graphs has not yet been successful, but some progress has been achieved recently. The aim of this note is to report on the successes so far.
PseudoRandom Graphs
 IN: MORE SETS, GRAPHS AND NUMBERS, BOLYAI SOCIETY MATHEMATICAL STUDIES 15
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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.