TY - JOUR

T1 - Testing satisfiability

AU - Alon, Noga

AU - Shapira, Asaf

N1 - Funding Information:
✩ A preliminary version of this paper appeared in the Proc. of the 13th Annual ACM–SIAM SODA, ACM Press, 2002, pp. 645–654. * Corresponding author. E-mail addresses: noga@math.tau.ac.il (N. Alon), asafico@math.tau.ac.il (A. Shapira). 1 Research supported in part by a USA–Israeli BSF grant, by the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. 2 Research supported by the Deutsch institute.

PY - 2003/7

Y1 - 2003/7

N2 - 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 ε-far from satisfiable, if one must remove at least εnk functions in order to make the set of remaining functions satisfiable. Our main result is that if Φ is ε-far from satisfiable, then most of the induced sets of functions, on sets of variables of size c(k, d)/ε2, 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 ε-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 ε-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.

AB - 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 ε-far from satisfiable, if one must remove at least εnk functions in order to make the set of remaining functions satisfiable. Our main result is that if Φ is ε-far from satisfiable, then most of the induced sets of functions, on sets of variables of size c(k, d)/ε2, 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 ε-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 ε-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.

KW - Hypergraph coloring

KW - Property testing

KW - Satisfiability

UR - http://www.scopus.com/inward/record.url?scp=0038310709&partnerID=8YFLogxK

U2 - 10.1016/S0196-6774(03)00019-1

DO - 10.1016/S0196-6774(03)00019-1

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AN - SCOPUS:0038310709

SN - 0196-6774

VL - 47

SP - 87

EP - 103

JO - Journal of Algorithms

JF - Journal of Algorithms

IS - 2

ER -