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

Abstract

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 Oum and 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 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].

Model Counting for CNF Formulas of Bounded Modular Treewidth

“…By combining these facts, we can conclude that modular incidence treewidth is more general than incidence treewidth. Incidence clique-width is known to be more general than signed incidence clique-width [10]. It is also more general than modular incidence treewidth.…”

“…This parameter is known to be more general than primal treewidth [13]. Again, #SAT on formulas of bounded incidence treewidth is linear-time tractable [10,23]. Clique-width is a graph invariant based on graph grammars [4].…”

“…The signed incidence clique-width of a formula corresponds to the signed clique-width of its signed incidence graph, which is obtained from the incidence graph by orientating edges so as to indicate positive or negative occurrences of variables. A polynomial-time algorithm for #SAT on formulas of bounded signed incidence clique-width is due to Fischer, Makowsky, and Ravve [10]. Clique-width is typically approximated by means of another parameter known as rank-width [20].…”

Solving #SAT using vertex covers

“…Whence, clique-width can be considered as the most general parameter considered so far. The fixed-parameter tractability of #SAT(cwd) follows from work of Courcelle, Makowsky, and Rotics [5] and that of Oum and Seymour [22], improved by Fischer, Makowsky, and Ravve [8]. Samer and Szeider [25] present dynamic programming algorithms for #SAT(tw) and #SAT(tw * ).…”

“…A similar time complexity can be achieved by restricting the treewidth of primal graphs and by dynamic programming on tree-decompositions; this approach is described by Gottlob, Scarcello, and Sideri [12] for SAT and can be extended to #SAT as explicated by Samer and Szeider [25]. Bounding the clique-width of directed incidence graphs yields larger classes of formulas for which #SAT is tractable: by combining Oum and Seymour's approximation algorithm for clique-width [22] with a general result of Courcelle, Makowsky, and Rotics [5] on counting problems expressible in a certain fragment of Monadic Second Order Logic, it can be shown that #SAT is fixed-parameter tractable for formulas of clique-width at most k. Fischer, Makowsky, and Ravve [8] improve the constants to obtain an algorithm that solves #SAT in time n O(1) O(f (k)) for formulas with n variables whose directed incidence graphs have clique-width k, where f is a simply exponential function. The latter result is more general than the results for bounded treewidth and branchwidth in the sense that every class of formulas with bounded treewidth or bounded branchwidth also has bounded clique-width; however, there are classes of formulas with bounded clique-width but unbounded treewidth and unbounded branchwidth; see Section 4.…”