Model Counting for CNF Formulas of Bounded Modular Treewidth

Abstract

The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for cliquewidth (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity. ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems

Solving #SAT and MAXSAT by Dynamic Programming

“…Both problems become tractable under certain structural restrictions obtained by bounding width parameters of graphs associated with formulas (Fischer, Makowsky, & Ravve, 2008;Ganian, Hlinený, & Obdrzálek, 2013;Samer & Szeider, 2010;Szeider, 2003). The work we present here is inspired by recent results in the work of Paulusma, Slivovsky, and Szeider (2013) and also in the work of Slivovsky and Szeider (2013) showing that #SAT is solvable in polynomial time when the incidence graph 1 I(F ) of the input formula F has bounded modular treewidth, and more strongly, bounded symmetric clique-width.…”

“…Since we are not solving a graph theoretic problem, expressing runtime by a graph theoretic parameter may be a limitation. Therefore, our strategy will be to develop a framework based on the following strategy (A) consider, for #SAT or MaxSAT, the amount of information needed to combine solutions to subproblems into global solutions, then (B) define the notion of good decompositions based on a parameter that minimizes this information, and then (C) design a dynamic programming algorithm along such a decomposition with runtime expressed by this parameter Both in the work of Paulusma et al (2013) and in that of Slivovsky and Szeider (2013) two assignments are considered to be equivalent if they satisfy the same set of clauses. When carrying out (A) for #SAT and MaxSAT this led us to the concept of ps-value of a CNF formula.…”

Revisiting Graph Width Measures for CNF-Encodings

“…From Theorem 2, we directly get lower bounds for the width measures studied in [30,35,16,38,34]. The first result considers the parameters with respect to which SAT is fixed-parameter tractable.…”

A Linear Time Algorithm for Computing #2SAT for Outerplanar 2-CNF Formulas

“…Mientras que para creación del árbol, se recorre cada entrada en la tabla, lo que requiere también del orden operaciones básicas. Finalmente, el recorrido del árbol es en postorden, y mientras se visitan los nodos del árbol, se va también visitando cada una de las aristas (cláusulas) de la gráfica de restricciones, al mismo tiempo que se van aplicando las recurrencias (1) y (2). Este último proceso nos genera del orden de ( + ) operaciones básicas.…”

“…# ( ) pertenece a la clase # -Completo aún cuando este en 2-FNC, este último denotado como #2 [1]. Aunque el problema #2 es # -Completo, existen instancias que se pueden resolver en tiempo polinomial [2]. Por ejemplo, si la gráfica que representa la fórmula es acíclica, entonces #2 puede resolverse en tiempo polinomial.…”