What makes normalized weighted satisfiability tractable

We consider the weighted antimonotone and the weighted monotone 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 antimonotone and monotone propositional formulas (including weighted anitmonoone/monotone cnf-sat) in a natural way, and serve as the canonical problems in the definition of the parameterized complexity hierarchy. We characterize the parameterized complexity of wsat−[t] and wsat[t] with respect to the genus of the circuit. For wsat−[t], which is W [t]-complete for odd t and W [t− 1]-complete for even t, the characterization is precise: We show that wsat−[t] is fixed-parameter tractable (FPT) if the genus of the circuit is n (n is the number of the variables in the circuit), and that it has the same W -hardness as the general wsat−[t] problem (i.e., with no restriction on the genus) if the genus is n. For wsat[2] (i.e., weighted monotone cnf-sat), which is W [2]-complete, the characterization is also precise: We show that wsat[2] is FPT if the genus is n and W [2]-complete if the genus is n. For wsat[t] where t > 2, which is W [t]-complete for even t and W [t − 1]-complete for odd t, we show that it is FPT if the genus is O(√logn), and that it has the same W -hardness as the general wsat[t] problem if the genus is n.