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33
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 47 (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.
Every minorclosed property of sparse graphs is testable
, 2007
"... Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a sim ..."
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Cited by 25 (3 self)
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Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a similar result is proved for any minorclosed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outerplanar, seriesparallel, bounded genus, bounded treewidth and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. 1
SOLVING VARIATIONAL INEQUALITIES WITH STOCHASTIC MIRRORPROX ALGORITHM
, 2008
"... Abstract. In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a n ..."
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Cited by 23 (5 self)
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Abstract. In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic MirrorProx (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters. We apply the SMP algorithm to Stochastic composite minimization and describe particular applications to Stochastic Semidefinite Feasability problem and Eigenvalue minimization. Key words. Nash variational inequalities, stochastic convexconcave saddlepoint problem, large scale stochastic approximation, reduced complexity algorithms for convex optimization AMS subject classifications. 90C15, 65K10, 90C47 1. Introduction. Let
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,
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 19 (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 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 15 (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
A unified framework for testing linearinvariant properties
 In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science
, 2010
"... In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and F ..."
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Cited by 10 (5 self)
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In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linearinvariant properties, drawing on techniques from additive combinatorics and from graph theory. Our main contributions here are the following: 1. We introduce a simple combinatorial condition, which we call subspaceheredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area. 2. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with onesided error, thus addressing a challenge posed by Sudan recently. 3. We introduce a new technique for proving the testability of Boolean functions. Using it, we verify a special case of the conjecture. Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.
Testing the expansion of a graph
, 2007
"... We study the problem of testing the expansion of graphs with bounded degree d in sublinear time. A graph is said to be an αexpander if every vertex set U ⊂ V of size at most 1V  has a neigh2 borhood of size at least αU. We show that the algorithm proposed by Goldreich and Ron [9] (ECCC2000) f ..."
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Cited by 9 (1 self)
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We study the problem of testing the expansion of graphs with bounded degree d in sublinear time. A graph is said to be an αexpander if every vertex set U ⊂ V of size at most 1V  has a neigh2 borhood of size at least αU. We show that the algorithm proposed by Goldreich and Ron [9] (ECCC2000) for testing the expansion of a graph distinguishes with high probability between αexpanders of degree bound d and graphs which are ɛfar from having expansion at least Ω(α2). This improves a recent result of Czumaj and Sohler [3] (FOCS07) who showed that this algorithm can distinguish between αexpanders of degree bound d and graphs which are ɛfar from having expansion at least Ω(α2 / log n). It also improves a recent result of Kale and Seshadhri [12] (ECCC2007) who showed that this algorithm can distinguish between αexpanders and graphs which are ɛfar from having expansion at least Ω(α2) with twice the maximum degree. Our methods combine the techniques of [3], [9] and [12]. 1
Sublineartime approximation for clustering via random sampling
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP’04
, 2004
"... Abstract. In this paper we present a novel analysis of a random sampling approach for three clustering problems in metric spaces: kmedian, minsum kclustering, and balanced kmedian. For all these problems we consider the following simple sampling scheme: select a small sample set of points unifor ..."
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Cited by 8 (1 self)
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Abstract. In this paper we present a novel analysis of a random sampling approach for three clustering problems in metric spaces: kmedian, minsum kclustering, and balanced kmedian. For all these problems we consider the following simple sampling scheme: select a small sample set of points uniformly at random from V and then run some approximation algorithm on this sample set to compute an approximation of the best possible clustering of this set. Our main technical contribution is a significantly strengthened analysis of the approximation guarantee by this scheme for the clustering problems. The main motivation behind our analyses was to design sublineartime algorithms for clustering problems. Our second contribution is the development of new approximation algorithms for the aforementioned clustering problems. Using our random sampling approach we obtain for the first time approximation algorithms that have the running time independent of the input size, and depending on k and the diameter of the metric space only. 1
Testing monotone continuous distributions on highdimensional real cubes
 In Proceedings of 21st ACMSIAM Symposium on Discrete Algorithms
, 2010
"... We study the task of testing properties of probability distributions. We consider a scenario in which we have access to independent samples of an unknown distribution ..."
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Cited by 8 (0 self)
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We study the task of testing properties of probability distributions. We consider a scenario in which we have access to independent samples of an unknown distribution