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

We study random instances of the weighted d-CNF satisfiability problem (WEIGHTED dSAT), a generic W[1]-complete problem. A random instance of the problem consists of a fixed parameter k and a random d-CNF formula F 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 − 1 clauses that contain at least one negated literals. We show that random instances of WEIGHTED d-SAT can be solved in O(kn + n)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 F 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 F k,d , provides an almost complete characterization of the typical-case behavior of random instances of WEIGHTED d-SAT.