Counting truth assignments of formulas of bounded tree-width or clique-width

We study algorithms for ]SAT and its generalized version ]GENSAT, the problem of computing the number of satisfying assignments of a set of propositional clauses . For this purpose we consider the clauses given by their incidence graph, a signed bipartite graph SI( ), and its derived graphs I( ) and P ( ). It is well known, that, given a graph of tree-width k, a k-tree decomposition can be found in polynomial time. Very recently S. Oum and P. Seymour have shown that, given a graph of clique-width k, a (2 3k+2 1)-parse tree witnessing clique-width can be found in polynomial time. In this paper we present an algorithm for ]GENSAT for formulas of bounded tree-width k which runs in time 4 k (n + n 2 log 2 (n)), where n is the size of the input. The main ingredient of the algorithm is a splitting formula for the number of satisfying assignments for a set of clauses where the incidence graph I( ) is a union of two graphs G 1 and G 2 with a shared induced subgraph H of size at most k. We also present analogue improvements for algorithms for formulas of bounded clique-width which are given together with their derivation. This improves considerably results for ]SAT, and hence also for SAT, previously obtained by Courcelle, Makowsky and Rotics, [CMR01].