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 0.2325n ) and O(2 0.1379n ), respectively. The previously best algorithms have running times O(2 0.2441n ) for Exact Satisfiability (Methods Oper. Res. 43 (1981) 419-431) and O(2 0.1626n ) 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. (J.M. Byskov), bolette@brics.dk (B.A. Madsen), skjernaa@brics.dk (B. Skjernaa).

The Exact Satisfiability problem is to determine if a CNFformula 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 0.2325n ) and O(2 0.1379n ), respectively. The previously best algorithms have running times O(2 0.2441n ) for Exact Satisfiability (Monien, Speckenmeyer and Vornberger (1981)) and O(2 0.1626n ) for Exact 3-Satisfiability (Kulikov and independently Porschen, Randerath and Speckenmeyer (2002)). 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.

Inspired by the Maximum Satisfiability and Exact Satisfiability problems we present two Maximum Exact Satisfiability problems. The first problem called Maximum Exact Satisfiability is: given a formula in conjunctive normal form and an integer k, is there an assignment to all variables in the formula such that at least k clauses have exactly one true literal. The second problem called Restricted Maximum Exact Satisfiability has the further restriction that no clause is allowed to have more than one true literal. Both problems are proved NP-complete restricted to the versions where each clause contains at most two literals. In fact Maximum Exact Satisfiability is a generalisation of the well-known NP-complete problem MaxCut. We present an exact algorithm for Maximum Exact Satisfiability where each clause contains at most two literals with time complexity O(poly(L)·2 m/4 ), where m is the number of clauses and L is the length of the formula. For the second version we give an algorithm with time complexity O(poly(L) · 1.324718 n ), where n is the number of variables. We note that when restricted to the versions where each clause contains exactly two literals and there are no negations both problems are fixed parameter tractable. It is an open question if this is also the case for the general problems.

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We give an algorithm for Exact Satisfiability with polynomial space usage and a time bound of poly(L) . m!, where m is the number of clauses and L is the length of the formula. Skjernaa has given an algorithm for Exact Satisfiability with time bound poly(L) . 2^m but using exponential space. We leave the following problem open: Is there an algorithm for Exact Satisfiability using only polynomial space with a time bound of c^m, where c is a constant and m is the number of clauses?