When is weighted satisfiability in FPT?∗

We consider the weighted monotone and antimonotone satisfiability problems on normalized circuits of depth at most t ≥ 2, abbreviated wsat[t] and wsat−[t], respectively. These problems model the weighted satisfiability of monotone and antimonotone propositional formulas (including weighted monotone/antimonotone cnf-sat) in a natural way, and serve as the canonical problems in the definition of the parameterized complexity hierarchy. In particular, wsat[t] (t ≥ 2) is W[t]-complete for even t and W[t − 1]-complete for odd t, and wsat−[t] (t ≥ 2) is W[t]-complete for odd t and W[t− 1]-complete for even t. Moreover, several well-studied problems, including important graph problems, can be modeled as wsat[t] and wsat−[t] problems in a straightforward manner. We study the parameterized complexity of wsat−[t] and wsat[t] 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, where n is the number of the variables in the circuit. For wsat[2] (i.e., weighted monotone cnf-sat), which is W[2]-complete, we also give FPT-algorithms when the genus is n. For wsat[t] where t ≥ 3, we give FPT-algorithms when the genus is o(log (n)). We also show that both wsat−[t] and wsat[t] on circuits of genus n 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[2] 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.