New algorithms for Exact Satisfiability

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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(20.2325n) and O(20.1379n), respectively. The previously best algorithms have running times O(20.2441n) for Exact Satisfiability (Methods Oper. Res. 43 (1981) 419–431) and O(20.1626n) 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.

Keywords

Exact Satisfiability
Exact 3-Satisfiability
Exact solution
Branch-and-reduce algorithm
Exponential-time algorithm

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Partially supported by the Future and Emerging Technologies programme of the EU under contract number IST-1999-14186 (ALCOM-FT).

1

Part of this research was done while visiting School of Information & Computer Science, University of California, Irvine, USA.

2

Part of this research was done while visiting Max-Planck-Institut für Informatik, Saarbrücken, Germany.