What makes normalized weighted satisfiability tractable
Abstract
We consider the weighted antimonotone and the weighted monotone satisfiability problems on normalized circuits of depth at most $t \geq 2$, abbreviated {\sc wsat$^[t]$} and {\sc wsat$^+[t]$}, respectively. These problems model the weighted satisfiability of antimonotone and monotone propositional formulas (including weighted anitmonoone/monotone {\sc cnfsat}) in a natural way, and serve as the canonical problems in the definition of the parameterized complexity hierarchy. We characterize the parameterized complexity of {\sc wsat$^[t]$} and {\sc wsat$^+[t]$} with respect to the genus of the circuit. For {\sc wsat$^[t]$}, which is $W[t]$complete for odd $t$ and $W[t1]$complete for even $t$, the characterization is precise: We show that {\sc wsat$^[t]$} is fixedparameter tractable (FPT) if the genus of the circuit is $n^{o(1)}$ ($n$ is the number of the variables in the circuit), and that it has the same $W$hardness as the general {\sc wsat$^[t]$} problem (i.e., with no restriction on the genus) if the genus is $n^{O(1)}$. For {\sc wsat$^+[2]$} (i.e., weighted monotone {\sc cnfsat}), which is $W[2]$complete, the characterization is also precise: We show that {\sc wsat$^+[2]$} is FPT if the genus is $n^{o(1)}$ and $W[2]$complete if the genus is $n^{O(1)}$. For {\sc wsat$^+[t]$} where $t > 2$, which is $W[t]$complete for even $t$ and $W[t1]$complete for odd $t$, we show that it is FPT if the genus is $O(\sqrt{\log{n}})$, and that it has the same $W$hardness as the general {\sc wsat$^+[t]$} problem if the genus is $n^{O(1)}$.
 Publication:

arXiv eprints
 Pub Date:
 December 2011
 arXiv:
 arXiv:1112.1040
 Bibcode:
 2011arXiv1112.1040K
 Keywords:

 Computer Science  Computational Complexity