SAT-Problems and Reductions with Respect to the Number of Variables

Abstract

In many algorithms or heuristics for the SAT problem the number of distinct variables occurring in the formula seems to play a central role, e.g. in Davis-Putnam algorithms. For deterministic algorithms, Schiermeyer has proved that 3-SAT is solvable in deterministic time O({vert_bar} {Phi} {vert_bar} c{sup n}), where n is the number of variables of {Phi} and c = 1.579. In contrast, no such algorithm is known (for any c 1 Unique SAT is decidable in deterministic time O({vert_bar} {Phi} {vert_bar} c{sup n}) iff SAT is decidable in deterministic time O({vert_bar} {Phi} {vert_bar} c{sup n}), where n ismore » the number of variables. With slight modifications this holds for k-Unique SAT and k-SAT, too. Finally we present a proof that for each problem in NTIME(n) there is a polynomial reduction to SAT such that the number of variables in f({Phi}) is only O(n) improving Schnorr-Cook`s reduction with O(nlogn) variables.« less