New Algorithms for Exact Satisfiability

Abstract

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.

Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

“…Feder and Motwani [18] give a randomised linear space algorithm with running time O((χ/e) n ) (with high probability), and the running time of a recent algorithm by Angelsmark and Thapper [1] can be given as O((2 + log χ) n ), an asymptotic improvement over Christofides' result for all values of χ . Prior to the current paper, no polynomial space algorithm with running time O(c n ) was known for any constant c, an open problem observed in [6,28]. We provide two such algorithms, based on divide-and-conquer in time O (8.33 n …”

“…Although the two ideas used are far from new, their combination demonstrates a candidate for obtaining competitive upper bounds for some set cover problems other than through analyzing branch and bound algorithms like in [7,12,16,30]. Furthermore, the hard cases of the analysis in #XSAT in [12] and in Minimum Dominating Set in [16] are those when all sets are rather small.…”

“…Prior to the current paper, no polynomial space algorithm with running time c m l O (1) was known for any constant c, an open problem observed in [6,28].…”

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

“…time). The best known algorithm for Exact Satisfiability (no limit on clause length) has a running time of poly(L) · 2 0.2325n [2]. 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.…”

“…Various exact algorithms have been given for this problem [7,2]. So far all algorithms given for XSAT have been analysed using the number of variables as parameter (in the exponentially growing part of the running E-mail address: bolette@brics.dk (B.A.…”