When is weighted satisfiability FPT?

• Authors: Iyad A. Kanj;Ge Xia
• Venue: WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
• Year: 2013

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 no(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 no(1). For wsat+[t] where t3, 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 nΩ(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.