Exact 3-satisfiability is decidable in time O(2^0.16254n)

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).