When Is Weighted Satisfiability FPT?

Abstract

The weighted monotone and antimonotone satisfiability problems on normalized circuits, abbreviated wsat ⁺[t] and wsat ⁻[t], are canonical problems in the parameterized complexity theory. We study the parameterized complexity of wsat ⁻[t] and wsat ⁺[t], where t ≥ 2, with respect to the genus of the circuit. For wsat ⁻[t], we give a fixed-parameter tractable (FPT) algorithm when the genus of the circuit is n ^o(1), where n is the number of the variables in the circuit. For wsat ⁺[2] (i.e., weighted monotone cnf-sat) and wsat ⁺[3], which are both W[2]-complete, we also give FPT algorithms when the genus is n ^o(1). For wsat ⁺[t] where t > 3, we give FPT algorithms when the genus is $O(\sqrt{\log{n}})$. We also show that both wsat ⁻[t] and wsat ⁺[t] on circuits of genus n ^Ω(1) have the same W-hardness as the general wsat ⁺[t] and wsat ⁻[t] problem (i.e., with no restriction on the genus), thus drawing a precise map of the parameterized complexity of wsat ⁻[t], and of wsat ⁺[t], for t = 2,3, with respect to the genus of the underlying circuit.

As a byproduct of our results, we obtain, via standard parameterized reductions, tight results on the parameterized complexity of several problems with respect to the genus of the underlying graph.