Showing papers by "Asaf Shapira published in 2005"

A characterization of the (natural) graph properties testable with one-sided error

Abstract: 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 P has an oblivious one-sided error tester, if and only if P is (semi) hereditary. 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. These include the results of Goldreich et al., [1998] about testing k-colorability, the characterization of Goldreich and Trevisan [2001] of the graph-partition problems that are testable with 1-sided error, the induced vertex colorability properties of Alon et al., [2000], the induced edge colorability properties of Fischer [2001], a transformation from 2-sided to 1-sided error testing [Goldreich and Trevisan, 2001], as well as a recent result about testing monotone graph properties [Alon and Shapira, 2005]. 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 and more. None of these properties was previously known to be testable.

Every monotone graph property is testable

Abstract: 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 e. 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 Szemeredi'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 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 e -far from satisfying P, contains a subgraph of size depending on e only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is e-far from satisfying a (possibly infinite) set of graph properties P, then it is at least δ P e-far from satisfying one of the properties.

Additive approximation for edge-deletion problems

Abstract: A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E/sub P/'(G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. 1) For any /spl epsiv/ > 0 and any monotone property P, there is a deterministic algorithm, which given a graph G of size n, approximates E/sub P/'(G) in time O(n/sup 2/) to within an additive error of /spl epsiv/n/sup 2/. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E/sub P/'. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist; 1. If there is a bipartite graph that does not satisfy P, then there is a /spl delta/ > 0 for which it is possible to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta// in polynomial time. 2) On the other hand, if all bipartite graphs satisfy P, then for any /spl delta/ > 0 it is NP-hard to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta//. While the proof of (1) is simple, the proof of (2) requires several new ideas and involves tools from extremal graph theory together with spectral techniques. This approach may be useful for obtaining other hardness of approximation results. Interestingly, prior to this work it was not even known that computing E/sub P/' precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis [1981], who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E/sub P/' is NP-hard.

Homomorphisms in Graph Property Testing - A Survey

Abstract: Property-testers are fast randomized algorithms for distinguishing between graphs (and other combinatorial structures) satisfying a certain property, from those that are far from satisfying it In many cases one can design property-testers whose running time is in fact independent of the size of the input In this paper we survey some recent results on testing graph properties A common thread in all the results surveyed is that they rely heavily on the simple yet useful notion of graph homomorphism We mainly focus on the combinatorial aspects of property-testing

Linear equations, arithmetic progressions and hypergraph property testing

Abstract: For a fixed k-uniform hypergraph D (k-graph for short, k ≥ 3), we say that a k-graph H satisfies property PD (resp. P*D) if it contains no copy (resp. induced copy) of D. Our goal in this paper is to classify the k-graphs D for which there are property-testers for testing PD and P*D whose query complexity is polynomial in 1/e. For such k-graphs, we say that PD (or P*D) is easily testable.For P*D, we prove that aside from a single 3-graph, P*D is easily testable if and only ifD is a single k-edge. For large k, we obtain stronger lower bounds than those obtained for the general case on the query complexity of testing P*D for any D other than the single k-edge. These bounds are proved by applying a more sophisticated technique than the basic one that works for all k. These results extend and improve previous results about graphs [5] and k-graphs [18].For PD, we show that for any k-partite k-graph D, PD, is easily testable, by giving an efficient one-sided error-property tester, which improves the one obtained by [18]. We further prove a nearly matching lower bound on the query complexity of such a property-tester. Finally, we give a sufficient condition for inferring that PD is not easily testable. Though our results do not supply a complete characterization of the k-graphs for which PD is easily testable, they are a natural extension of the previous results about graphs [1].Our proofs combine results and arguments from additive number theory, linear algebra and extremal hypergraph theory. We also develop new techniques, which are of independent interest. The first is a construction of a dense set of integers, which does not contain a subset that satisfies a certain set of linear equations. The second is an algebraic construction of certain extremal hypergraphs. We demonstrate the applicability of this last construction by resolving several cases of an open problem raised by Brown, Erdos and Sos in 1973. These two techniques have already been applied in two recent subsequent papers [6], [27].

Linear Equations, Arithmetic Progressions and Hypergraph Property Testing

Abstract: For a fixed k-uniform hypergraph D (k-graph for short, k ≥ 3), we say that a k-graph H satisfies property PD (resp. P*D) if it contains no copy (resp. induced copy) of D. Our goal in this paper is to classify the k-graphs D for which there are property-testers for testing PD and P*D whose query complexity is polynomial in 1/e. For such k-graphs, we say that PD (or P*D) is easily testable.For P*D, we prove that aside from a single 3-graph, P*D is easily testable if and only ifD is a single k-edge. For large k, we obtain stronger lower bounds than those obtained for the general case on the query complexity of testing P*D for any D other than the single k-edge. These bounds are proved by applying a more sophisticated technique than the basic one that works for all k. These results extend and improve previous results about graphs [5] and k-graphs [18].For PD, we show that for any k-partite k-graph D, PD, is easily testable, by giving an efficient one-sided error-property tester, which improves the one obtained by [18]. We further prove a nearly matching lower bound on the query complexity of such a property-tester. Finally, we give a sufficient condition for inferring that PD is not easily testable. Though our results do not supply a complete characterization of the k-graphs for which PD is easily testable, they are a natural extension of the previous results about graphs [1].Our proofs combine results and arguments from additive number theory, linear algebra and extremal hypergraph theory. We also develop new techniques, which are of independent interest. The first is a construction of a dense set of integers, which does not contain a subset that satisfies a certain set of linear equations. The second is an algebraic construction of certain extremal hypergraphs. We demonstrate the applicability of this last construction by resolving several cases of an open problem raised by Brown, Erdos and Sos in 1973. These two techniques have already been applied in two recent subsequent papers [6], [27].

Homomorphisms in Graph Property Testing - A Survey Dedicated to Jaroslav Ne set ril on the occasion of his 60 th birthday

Abstract: Property-testers are fast randomized algorithms for distinguishing between graphs (and other combinatorial structures) satisfying a certain property, from those that are far from satisfying it. In many cases one can design property-testers whose running time is in fact independent of the size of the input. In this paper we survey some recent results on testing graph properties. A common thread in all the results surveyed is that they rely heavily on the simple yet useful notion of graph homomorphism. We mainly focus on the combinatorial aspects of property-testing.