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13
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.
Algorithmic and Analysis Techniques in Property Testing
"... Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform ..."
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Cited by 24 (3 self)
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Property testing algorithms are “ultra”efficient algorithms that decide whether a given object (e.g., a graph) has a certain property (e.g., bipartiteness), or is significantly different from any object that has the property. To this end property testing algorithms are given the ability to perform (local) queries to the input, though the decision they need to make usually concern properties with a global nature. In the last two decades, property testing algorithms have been designed for many types of objects and properties, amongst them, graph properties, algebraic properties, geometric properties, and more. In this article we survey results in property testing, where our emphasis is on common analysis and algorithmic techniques. Among the techniques surveyed are the following: • The selfcorrecting approach, which was mainly applied in the study of property testing of algebraic properties; • The enforce and test approach, which was applied quite extensively in the analysis of algorithms for testing graph properties (in the densegraphs model), as well as in other contexts;
Property Testing: A Learning Theory Perspective
"... Property testing deals with tasks where the goal is to distinguish between the case that an object (e.g., function or graph) has a prespecified property (e.g., the function is linear or the graph is bipartite) and the case that it differs significantly from any such object. The task should be perfor ..."
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Cited by 22 (4 self)
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Property testing deals with tasks where the goal is to distinguish between the case that an object (e.g., function or graph) has a prespecified property (e.g., the function is linear or the graph is bipartite) and the case that it differs significantly from any such object. The task should be performed by observing only a very small part of the object, in particular by querying the object, and the algorithm is allowed a small failure probability. One view of property testing is as a relaxation of learning the object (obtaining an approximate representation of the object). Thus property testing algorithms can serve as a preliminary step to learning. That is, they can be applied in order to select, very efficiently, what hypothesis class to use for learning. This survey takes the learningtheory point of view and focuses on results for testing properties of functions that are of interest to the learning theory community. In particular, we cover results for testing algebraic properties of functions such as linearity, testing properties defined by concise representations, such as having a small DNF representation, and more. 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,
Quantum property testing
 In Proceedings of 14th SODA
, 2003
"... Hein R"ohrig \Lambda \Lambda ..."
Testing expansion in boundeddegree graphs
 Proc. of FOCS 2007
"... We consider the problem of testing expansion in bounded degree graphs. We focus on the notion of vertexexpansion: an αexpander is a graph G = (V, E) in which every subset U ⊆ V of at most V /2 vertices has a neighborhood of size at least α · U. Our main result is that one can distinguish good ..."
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Cited by 14 (1 self)
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We consider the problem of testing expansion in bounded degree graphs. We focus on the notion of vertexexpansion: an αexpander is a graph G = (V, E) in which every subset U ⊆ V of at most V /2 vertices has a neighborhood of size at least α · U. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time Õ(√n). We prove that the property testing algorithm proposed by Goldreich and Ron with appropriately set parameters accepts every αexpander with probability at least 2, where α ∗ ( α = Θ 2 d2) and d is the maximum degree of the graphs. The algorithm assumes the log(n/ε) boundeddegree) graphs model with adjacency list graph representation and its running time is O and rejects every graph that is εfar from any α∗expander with probability at least 2 3 d 2 √ n log(n/ε) α 2 ε 3
Sublineartime algorithms
 In Oded Goldreich, editor, Property Testing, volume 6390 of Lecture Notes in Computer Science
, 2010
"... In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1 ..."
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Cited by 10 (2 self)
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In this paper we survey recent (up to end of 2009) advances in the area of sublineartime algorithms. 1
Testing hypergraph coloring
 Proc. of ICALP 2001
, 2001
"... Abstract. In this paper we initiate the study of testing properties of hypergraphs. The goal of property testing is to distinguish between the case whether a given object has a certain property or is “far away ” from the property. We prove that the fundamental problem of ℓcolorability of kuniform ..."
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Cited by 9 (1 self)
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Abstract. In this paper we initiate the study of testing properties of hypergraphs. The goal of property testing is to distinguish between the case whether a given object has a certain property or is “far away ” from the property. We prove that the fundamental problem of ℓcolorability of kuniform hypergraphs can be tested in time independent of the size of the hypergraph. We present a testing algorithm that examines only (k ℓ/ɛ) O(k) entries of the adjacency matrix of the input hypergraph, where ɛ is a distance parameter independent of the size of the hypergraph. Notice that this algorithm tests only a constant number of entries in the adjacency matrix provided that ℓ, k, and ɛ are constant. 1
Smoothed Analysis: Motivation and Discrete Models
 Proc. of WADS 2003
, 2003
"... Abstract. In smoothed analysis, one measures the complexity of algorithms assuming that their inputs are subject to small amounts of random noise. In an earlier work (Spielman and Teng, 2001), we introduced this analysis to explain the good practical behavior of the simplex algorithm. In this paper, ..."
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Cited by 8 (0 self)
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Abstract. In smoothed analysis, one measures the complexity of algorithms assuming that their inputs are subject to small amounts of random noise. In an earlier work (Spielman and Teng, 2001), we introduced this analysis to explain the good practical behavior of the simplex algorithm. In this paper, we provide further motivation for the smoothed analysis of algorithms, and develop models of noise suitable for analyzing the behavior of discrete algorithms. We then consider the smoothed complexities of testing some simple graph properties in these models. 1