A Fixed-Parameter Algorithm for Random Instances of Weighted d-CNF Satisfiability

We study random instances of the weighted $d$-CNF satisfiability problem (WEIGHTED $d$-SAT), a generic W[1]-complete problem. A random instance of the problem consists of a fixed parameter $k$ and a random $d$-CNF formula $\weicnf{n}{p}{k, d}$ generated as follows: for each subset of $d$ variables and with probability $p$, a clause over the $d$ variables is selected uniformly at random from among the $2^d - 1$ clauses that contain at least one negated literals. We show that random instances of WEIGHTED $d$-SAT can be solved in $O(k²ⁿ + n^{{O(1)})$-time} with high probability, indicating that typical instances of WEIGHTED $d$-SAT under this instance distribution are fixed-parameter tractable. The result also hold for random instances from the model $\weicnf{n}{p}{k,d}(d')$ where clauses containing less than $d' (1 < d' < d)$ negated literals are forbidden, and for random instances of the renormalized (miniaturized) version of WEIGHTED $d$-SAT in certain range of the random model's parameter $p(n)$. This, together with our previous results on the threshold behavior and the resolution complexity of unsatisfiable instances of $\weicnf{n}{p}{k, d}$, provides an almost complete characterization of the typical-case behavior of random instances of WEIGHTED $d$-SAT.