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 @S. For this purpose we consider the clauses given by their incidence graph, a signed bipartite graph SI(@S), and its derived graphs I(@S) and P(@S). It is well known, that, given a graph of tree-width k, a k-tree decomposition can be found in polynomial time. Very recently Oum and Seymour have shown that, given a graph of clique-width k, a (2^3^k^+^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 @S where the incidence graph I(@S) 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 considerably improves results for @?SAT, and hence also for SAT, previously obtained by Courcelle et al. [On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic, Discrete Appl. Math. 108 (1-2) (2001) 23-52].