An algorithm for Exact Satisfiability analysed with the number of clauses as parameter

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 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, where c is a constant and m is the number of clauses? Exact Satisfiability (XSAT) is the problem: given a formula F in conjunctive normal form, is there an assignment to all variables in F , such that exactly one literal in each clause is true? In this paper a formula F has m clauses and n variables. A literal is either a variable or the negation of a variable. The length of a formula L is the number of literals in the formula. XSAT is NP-complete even when restricted to clauses containing at most three literals and all variables occurring only unnegated [7], and various exact algorithms have been given for this problem [6, 5]. So far all algorithms given for XSAT have been analysed using the number of variables as parameter. The best known algorithm for Exact Satisfiability (no limit on clause length) has a running time of poly(L) · 20.2325n [5]. This algorithm (or a variant thereof) also gives a time bound in the number of literals, but no good time bound in the number of clauses is known. This is interestingly different from Satisfiability (no limit on clause length) for which good time bounds in the number of clauses have been proved (the ∗Basic Research in Computer Science (www.brics.dk), funded by the Danish National Research Foundation.