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28
Some 3CNF properties are hard to test
 In Proc. 35th ACM Symp. on Theory of Computing
, 2003
"... Abstract. For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of nbit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that th ..."
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Abstract. For a Boolean formula ϕ on n variables, the associated property Pϕ is the collection of nbit strings that satisfy ϕ. We study the query complexity of tests that distinguish (with high probability) between strings in Pϕ and strings that are far from Pϕ in Hamming distance. We prove that there are 3CNF formulae (with O(n) clauses) such that testing for the associated property requires Ω(n) queries, even with adaptive tests. This contrasts with 2CNF formulae, whose associated properties are always testable with O ( √ n) queries [E. Fischer et al., Monotonicity testing over general poset domains, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 474–483]. Notice that for every negative instance (i.e., an assignment that does not satisfy ϕ) there are three bit queries that witness this fact. Nevertheless, finding such a short witness requires reading a constant fraction of the input, even when the input is very far from satisfying the formula that is associated with the property. A property is linear if its elements form a linear space. We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include the following observations which are of independent interest: 1. In the context of testing for linear properties, adaptive twosided error tests have no more power than nonadaptive onesided error tests. Moreover, without loss of generality, any test for a linear property is a linear test. A linear test verifies that a portion of the input satisfies a set of linear constraints, which define the property, and rejects if and only if it finds a falsified constraint. A linear test is by definition nonadaptive and, when applied to linear properties, has a onesided error. 2. Random low density parity check codes (which are known to have linear distance and constant rate) are not locally testable. In fact, testing such a code of length n requires Ω(n) queries.
Algebraic Property Testing: The Role of Invariance
, 2007
"... We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider Flinear properties that are i ..."
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Cited by 40 (16 self)
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We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider Flinear properties that are invariant under linear transformations of the domain and prove that an O(1)local “characterization ” is a necessary and sufficient condition for O(1)local testability when K  = O(1). (A local characterization of a property is a definition of a property in terms of local constraints satisfied by functions exhibiting a property.) For the subclass of properties that are invariant under affine transformations of the domain, we prove that the existence of a single O(1)local constraint implies O(1)local testability. These results generalize and extend the class of algebraic properties, most notably linearity and lowdegreeness, that were previously known to be testable. In particular, the extensions include properties satisfied by functions of degree linear in n that turn out to be O(1)locally testable. Our results are proved by introducing a new notion that we term “formal characterizations”. Roughly this corresponds to characterizations that are given by a single local constraint and its permutations under linear transformations of the domain. Our main testing result shows that local formal characterizations
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 38 (7 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
Tight bounds for testing bipartiteness in general graphs
 SICOMP
"... In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs, and one most suitable for boundeddegree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant c ..."
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Cited by 38 (13 self)
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In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs, and one most suitable for boundeddegree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant complexity, while the complexity of testing boundeddegree graphs is ˜ Θ ( √ n), where n is the number of vertices in the graph (and ˜ Θ(f(n)) means Θ(f(n) · polylog(f(n)))). Thus there is a large gap between the complexity of testing in the two cases. In this work we bridge the gap described above. In particular, we study the problem of testing bipartiteness in a model that is suitable for all densities. We present an algorithm whose complexity is Õ(min( √ n, n 2 /m)) where m is the number of edges in the graph, and match it with an almost tight lower bound. This work is part of the author’s Ph.D. thesis prepared at Tel Aviv University under the supervision of Prof.
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 37 (6 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;
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
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,
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
Introduction to testing graph properties
 In Property Testing
, 2010
"... Abstract. The aim of this article is to introduce the reader to the study of testing graph properties, while focusing on the main models and issues involved. No attempt is made to provide a comprehensive survey of this ..."
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Abstract. The aim of this article is to introduce the reader to the study of testing graph properties, while focusing on the main models and issues involved. No attempt is made to provide a comprehensive survey of this