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?

For many problems, the investigation of their parameterized complexity provides an interesting and useful point of view. The most obvious natural parameterization for the maximum satisfiability problem—the number of satisfiable clauses—makes little sense, because at least half of the clauses can be satisfied in any formula. We look at two optimization variants of the exact satisfiability problem, where a clause is only said to be fulfilled iff exactly one of its literals is set to true. Interestingly, these variants behave quite differently. In the case of ResMaxExactSAT, where over-satisfied clauses are entirely forbidden, we show fixed parameter tractability. On the other hand, if we choose to ignore over-satisfied clauses, the MaxExactSAT problem is obtained. Surprisingly, it is W[1]-complete. Still, restricted variants of the problem turn out to be tractable.

An Algorithm for Exact Satisfiability Analysed with the Number of Clauses as Parameter

On the parameterized complexity of exact satisfiability problems

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An Algorithm for Exact Satisfiability Analysed with the Number of Clauses as Parameter

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On the parameterized complexity of exact satisfiability problems

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