Testing Hypergraph Coloring

Shapira

2003

Journal of Algorithms

Let Φ be a set of general boolean functions on n variables, such that each function depends on exactly k variables, and each variable can take a value from [1, d]. We say that Φ is -far from satisfiable, if one must remove at least n k functions in order to make the set of remaining functions satisfiable. Our main result is that if Φ is -far from satisfiable, then most of the induced sets of functions, on sets of variables of size c(k, d)/ 2 , are not satisfiable, where c(k, d) depends only on k and d. Using the above claim, we obtain similar results for k-SAT and k-NAEQ-SAT.Assume we relax the decision problem of whether an instance of one of the above mentioned problems is satisfiable or not, to the problem of deciding whether an instance is satisfiable or -far from satisfiable. While the above decision problems are NP-hard, our result implies that we can solve their relaxed versions, that is, distinguishing between satisfiable and -far from satisfiable instances, in randomized constant time.From the above result we obtain as a special case, previous results of Alon and Krivelevich, and of Czumaj and Sohler, concerning testing of graphs and hypergraphs colorability. We also discuss the difference between testing with one-sided and two-sided error.

Section: Applicationssupporting

confidence: 70%

“…In section 5, we explain briefly how to slightly improve the dependence on k an d in this last result of [10].…”

Section: For Every Fixed K and D And Sufficiently Smallmentioning

confidence: 99%

See 1 more Smart Citation

Testing satisfiability

Shapira

2003

Journal of Algorithms

“…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

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

Shapira

2003

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.

“…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

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

Véga

Kannan

et al. 2002

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.