Testing satistiability

Let Phi 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 Phi is is an element of-far from satisfiable, if one must remove at least is an element ofn(k) functions in order to make the set of remaining functions satisfiable. Our main result is that if Phi is is an element of-far from satisfiable, then most of the induced sets of functions, on sets of variables of size c(k, d)/is an element of(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 is an element of-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 is an element of-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. (C) 2003 Elsevier Science (USA). All rights reserved.