Exact 3-Satisfiability Is Decidable in Time O(20.16254n )

Abstract

Let F = C 1 ∧ · · · ∧ C m be a Boolean formula in conjunctive normal form over a set V of n propositional variables, s.t. each clause C i contains at most three literals l over V . Solving the problem exact 3-satisfiability (X3SAT) for F means to decide whether there is a truth assignment setting exactly one literal in each clause of F to true (1). As is well known X3SAT is NP-complete [6]. By exploiting a perfect matching reduction we prove that X3SAT is deterministically decidable in time O(2 0.18674n ). Thereby we improve a result in [2,3] stating X3SAT ∈ O(2 0.2072n ) and a bound of O(2 0.200002n ) for the corresponding enumeration problem #X3SAT stated in a preprint [1]. After that by a more involved deterministic case analysis we are able to show that X3SAT ∈ O(2 0.16254n ).

On variable-weighted exact satisfiability problems

“…The NP-hard optimization variants MINW-X3SAT, resp. MAXW-X3SAT, can be solved in O(2 0.16254n ) time [3,4] via techniques provided here and in [18]. However, it is left for future work to achieve for the optimization variants of X3SAT the up to now best bound of O(2 0.1379n ) for decision as obtained in [6].…”

New algorithms for Exact Satisfiability

“…All other assignments satisfying the new formula can be changed to satisfy the original one by setting x 1 to false and choosing the value of y 3 such that (x 2 , y 1 , y 3 ) is satisfied. By using this reduction, both x 1 and y 3 are removed, so we get a branching vector of at least (7,3). The third clause with x 1 contains none of the y i 's.…”

“…The previously best algorithms are by Porschen, Randerath and Speckenmeyer [7] and Kulikov [3] and have branching vector (9;4) corresponding to a running time of (2 0.1626n ) Extra reductions. For X3SAT we have some extra reductions which are only needed to remove certain cycles.…”

“…8, we get a branching vector of at least (2a + b + 1, 2b + a + 1) from the above clauses. If a + b 5, this yields a branching vector of at least (11, 6), (10, 7) or (9,8). For variables occurring fewer times, we also need to consider the other clauses in which the y's and y 's occur.…”

“…The first was by Drori and Peleg [2] and runs in time O (2 0.2072n ). This was improved by Kulikov [3] and independently Porschen et al [7] in 2002 to obtain a running time of O(2 0.1626n ). Dahllöf et al [1] have an alogorithm which they claim to run in time O(2 0.1532n ), but there is an error in the paper.…”