Asaf Shapira

A combinatorial characterization of the testable graph properties: it's all about regularity

A common thread in recent results concerning the testing of dense graphs is the use of Szemerédis 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 Szemerédi-partition 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 Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron [25] that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable.

Testing subgraphs in directed graphs

Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least en2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f(e, H)nh copies of H. This is proved by establishing a directed version of Szemeredis regularity lemma, and implies that for every H there is a one-sided error property tester whose query complexity is bounded by a function of e only for testing the property PH of being H-free.As is common with applications of the undirected regularity lemma, here too the function 1/f(e,H) is an extremely fast growing function in e. We therefore further prove a precise characterization of all the digraphs H, for which f(e,H) has a polynomial dependency on e. This implies a characterization of all the digraphs H, for which the property of being H-free has a one-sided error property tester whose query complexity is polynomial in 1/e. We further show that the same characterization also applies to two-sided error property testers as well. A special case of this result settles an open problem raised by the first author in (Alon, Proceedings of the 42nd IEEE FOCS, IEEE, New York, 2001, pp. 434-441). Interestingly, it turns out that if PH has a polynomial query complexity, then there is a two-sided e-tester for PH that samples only O(1/e) vertices, whereas any one-sided tester for PH makes at least (1/e)d/2 queries, where d is the average degree of H. We also show that the complexity of deciding if for a given directed graph H, PH has a polynomial query complexity, is NP-complete, marking an interesting distinction from the case of undirected graphs.For some special cases of directed graphs H, we describe very efficient one-sided error property-testers for testing PH. As a consequence we conclude that when H is an undirected bipartite graph, we can give a one-sided error property tester with query complexity O((1/e)h/2), improving the previously known upper bound O((1/e)h2). The proofs combine combinatorial, graph theoretic and probabilistic arguments with results from additive number theory.

Every monotone graph property is testable

A graph property is called monotone if it is closed under taking (not necessarily induced) subgraphs (or, equivalently, if it is closed under removal of edges and vertices). Many monotone graph properties are some of the most well-studied 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 one-sided error, and with query complexity depending only on ε. This result unifies several previous results in the area of property testing, and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerédis 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 one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph-theory, is that for any monotone graph property P, any graph that is ε -far from satisfying P, contains a subgraph of size depending on ε only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is ε-far from satisfying a (possibly infinite) set of graph properties P, then it is at least δ P ε-far from satisfying one of the properties.

A Characterization of the (Natural) Graph Properties Testable with One-Sided Error

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property testing. Our main result in this paper is a solution of an important special case of this general problem: Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property

A Combinatorial Characterization of the Testable Graph Properties: It's All About Regularity

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Every Monotone Graph Property Is Testable

has an oblivious one-sided error tester if and only if

Every minor-closed property of sparse graphs is testable

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A Characterization of Easily Testable Induced Subgraphs

is semihereditary. We stress that any “natural” property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the natural graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. More importantly, as a special case of our main result, we infer that some of the most well-studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well-known graph properties of being perfect, chordal, interval, comparability, permutation, and more. None of these properties was previously known to be testable.

Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs

A common thread in all of the recent results concerning the testing of dense graphs is the use of Szemeredis 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 Szemeredi-partition 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

Green's conjecture and testing linear-invariant properties

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