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.
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Kanj, I.A., Xia, G. (2013). When Is Weighted Satisfiability FPT?. In: Dehne, F., Solis-Oba, R., Sack, JR. (eds) Algorithms and Data Structures. WADS 2013. Lecture Notes in Computer Science, vol 8037. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40104-6_39
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DOI: https://doi.org/10.1007/978-3-642-40104-6_39
Publisher Name: Springer, Berlin, Heidelberg
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