Testing satisfiability

Alon, Noga; Shapira, A.

doi:10.1016/s0196-6774(03)00019-1

Cited by 28 publications

(23 citation statements)

References 17 publications

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“…Our method here improves this result and shows that a sample of size O((1/ ) min(s,t) ) suffices. This nearly matches a lower bound of Ω((1/ ) min(s,t)/2 ) which follows by considering an appropriate random graph (see the full version of [9]. )…”

Section: Proposition 81supporting

confidence: 78%

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Testing subgraphs in directed graphs

Alon¹,

Shapira²

2003

Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing - STOC '03

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Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least n 2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f ( , H)n h copies of H. This is proved by establishing a directed version of Szemerédi's 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 only for testing the property P H of being H-free.As is common with applications of the undirected regularity lemma, here too the function 1/f ( , H) is an extremely fast growing function in . We therefore further prove a precise characterization of all the digraphs H, for which f ( , H) has a polynomial dependency on . 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/ . 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 [1]. Interestingly, it turns out that if P H has a polynomial query complexity, then there is a two-sided -tester for P H that samples only O(1/ ) vertices, whereas any one-sided tester for P H makes at least (1/ ) 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, P H has a polynomial query complexity, is N P -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 propertytesters for testing P H . 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/ ) h/2 ), improving the previously known upper bound of O((1/ ) h 2 ). The proofs combine combinatorial, graph theoretic and probabilistic arguments with results from additive number theory.

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Section: Proposition 81supporting

confidence: 78%

“…The notion of property testing has been investigated in other areas as well, including the context of regular languages, [6], functions [23], [9], [3], computational geometry [18], [4] as well as graph and hypergraph coloring [17], [9], [15]. See [31] and [22] for surveys on the topic.…”

Section: Related Workmentioning

confidence: 99%

Testing subgraphs in directed graphs

Alon¹,

Shapira²

2003

Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing - STOC '03

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“…Their results relate the optimal solution value of the whole problem to a complicated function of the random sub-problems like [7] (see also [7], [5] and [2] for higher dimensional cases, or for cases in which our only objective is to decide if we can satisfy almost all constraints). Thus they differ from our new uniform method.…”

Section: Introductionmentioning

confidence: 99%

Random sampling and approximation of MAX-CSP problems

Alon¹,

Véga²,

Kannan³

et al. 2002

Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing - STOC '02

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We present a new efficient sampling method for approximating r-dimensional Maximum Constraint Satisfaction Problems, MAX-rCSP, on n variables up to an additive error n r . We prove a new general paradigm in that it suffices, for a given set of constraints, to pick a small uniformly random subset of its variables, and the optimum value of the subsystem induced on these variables gives (after a direct normalization and with high probability) an approximation to the optimum of the whole system up to an additive error of n r . Our method gives for the first time a polynomial in −1 bound on the sample size necessary to carry out the above approximation. Moreover, this bound is independent in the exponent on the dimension r. The above method gives a completely uniform sampling technique for all the MAX-rCSP problems, and improves the best known sample bounds for the low dimensional problems, like MAX-CUT.The method of solution depends on a new result on the cut norm of random subarrays, and a new sampling technique for high dimensional linear programs. This method could be also of independent interest.

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“…Item (4) in Lemma 2.7 states that each U i contains at least n/S( ) vertices. Also, by (6), and by the monotonicity properties of M 2.3 discussed in Comment 2.4, we have for any 1 ≤ i ≤ k…”

Section: If For Some I < J We Have |D(vmentioning

confidence: 76%

Every monotone graph property is testable

Alon

Shapira

2005

Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing

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

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Testing satisfiability

Cited by 28 publications

References 17 publications

Testing subgraphs in directed graphs

Testing subgraphs in directed graphs

Random sampling and approximation of MAX-CSP problems

Every monotone graph property is testable

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