On variable-weighted exact satisfiability problems

Abstract

We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O( C 2 0.2441n ) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n 2 · C + 2 0.40567n ) time. In recent years only the unweighted counterparts of these problems have been studied [8,9,15].

Algorithms for Variable-Weighted 2-SAT and Dual Problems

“…Next, we state a basic proposition relating bijections between Π-model spaces to bijections between weighted Π-model spaces, where Π ∈ {SAT, XSAT, NAESAT}. This result is a slight generalization of the same result restricted to weighted XSAT shown in [13,15]:…”

“…The first assertion slightly generalizes Lemma 7 in [15] restricted to weighted XSAT, and enabling us to reduce maximum weight problems to minimum weight problems in specific cases: Proof. Let C ⊆ CNF be a fixed formula class for which A is an algorithm solving MINW-Π for M -weighted input instances C ∈ C. We claim that T Π min (C, w) = T Π max (C, −w).…”

“…Lemma 6 ( [13,15]). For C ∈ CNF with w : V (C) → R, holds T XSAT = ∅ if there is c ∈ C containing more than one complemented pairs.…”

“…The main variants of satisfiability, namely exact satisfiability and not-all-equal satisfiability for weighted 2-CNF have been shown to be linear time solvable in [12]. Moreover, the optimization cases of XSAT for variable weighted formulas has been shown to be solvable in time O(2 0.2441n ) for arbitrary CNF [13,15], respectively, in time O(2 0.16254n ) restricted to 3-CNF [10,11]. Exactly solving NAESAT for arbitrary CNF formulas (or restricted to 3-CNF) exactly in less than the trivial 2 n steps remains an open problem for the decision problem as well as for its NP-hard variable-weighted optimization variants.…”

“…Exactly solving NAESAT for arbitrary CNF formulas (or restricted to 3-CNF) exactly in less than the trivial 2 n steps remains an open problem for the decision problem as well as for its NP-hard variable-weighted optimization variants. Recently also counting versions of weighted SAT have been considered: #XSAT for variable weighted CNF formulas can be solved in time O(n 2 · C +2 0.40567·n ) as shown in [14,15]. Fürer et al [5] provided an algorithm for counting all maximum weight solutions of SAT for variable weighted 2-CNF formulas.…”