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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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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,
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"... Property testing problems are relaxations of decision problems. A property testing algorithm (referred to as a testing algorithm or tester) has to decide if a given object has a prespeci ed property or is ɛfar from the property (for a given distance parameter ɛ, and for a prespeci ed distance measu ..."
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Property testing problems are relaxations of decision problems. A property testing algorithm (referred to as a testing algorithm or tester) has to decide if a given object has a prespeci ed property or is ɛfar from the property (for a given distance parameter ɛ, and for a prespeci ed distance measure). The tester is given query access to the input, and is required to run in sublinear time. In this thesis we focus on testing properties of directed graphs (digraphs). In particular we present the following results (where n is the number of vertices in the graph, d is the maximum degree, and davg is the average degree): 1. We present a testing algorithm for the property of Eulerianity in boundeddegree digraphs, which runs in time 1 Õ(1/ɛ). For unboundeddegree digraphs we show a lower bound of Ω ( √ n/ɛ), and give a testing algorithm that runs in time Õ( √ n/ɛ 3/2). 2. We consider the property of kedgeconnectivity in digraphs and present testing algorithms for both boundeddegree digraphs and unboundeddegree digraphs, that run in time
The Classification Problem in Relational Property Testing
, 2009
"... In property testing, we desire to distinguish between objects that have a given property and objects that are far from the property by examining only a small, randomly selected portion of the objects. Property testing arose in the study of formal verification, however much of the recent work has bee ..."
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In property testing, we desire to distinguish between objects that have a given property and objects that are far from the property by examining only a small, randomly selected portion of the objects. Property testing arose in the study of formal verification, however much of the recent work has been focused on testing graph properties. In this thesis we introduce a generalization of property testing which we call relational property testing. Because property testers examine only a very small portion of their inputs, there are potential applications to efficiently testing properties of massive structures. Relational databases provide perhaps the most obvious example of such massive structures, and our framework is a natural way to characterize this problem. We introduce a number of variations of our generalization and prove the relationships between them. The second major topic of this thesis is the classification problem for testability. Using the general framework developed in previous chapters, we consider the testability of various syntactic fragments of firstorder logic. This problem is inspired by the classical problem for decidability. We compare the current classi cation for testability with