New algorithms for Exact Satisfiability

Abstract

The Exact Satisfiability problem is to determine if a CNF-formula has a truth assignment satisfying exactly one literal in each clause; Exact 3-Satisfiability is the version in which each clause contains at most three literals. In this paper, we present algorithms for Exact Satisfiability and Exact 3-Satisfiability running in time O ( 2 0.2325 n ) and O ( 2 0.1379 n ) , respectively. The previously best algorithms have running times O ( 2 0.2441 n ) for Exact Satisfiability (Methods Oper. Res. 43 (1981) 419–431) and O ( 2 0.1626 n ) for Exact 3-Satisfiability (Annals of Mathematics and Artificial Intelligence 43 (1) (2005) 173–193 and Zapiski nauchnyh seminarov POMI 293 (2002) 118–128). We extend the case analyses of these papers and observe that a formula not satisfying any of our cases has a small number of variables, for which we can try all possible truth assignments and for each such assignment solve the remaining part of the formula in polynomial time.