A bottom-up algorithm for solving ♯2SAT

Counting models for a two conjunctive formula (2-CF) $F$, a problem known as $\sharp $2Sat, is a classic $\sharp $P complete problem. Given a 2-CF $F$ as input, its constraint graph $G$ is built. If $G$ is acyclic, then $\sharp $2Sat($F$) can be computed efficiently. In this paper, we address the case when $G$ has cycles. When $G$ is cyclic, we propose a decomposition on the constraint graph $G$ that allows the computation of $\sharp $2Sat($F$) in incremental way. Let $T$ be a cactus graph of $G$ containing a maximal number of independent cycles, and let $\overline{T}=(E(G)-E(T))$ be a subset of frond edges from $G$. The clauses in $\overline{T}$ are ordered in connected components $\{K_1, \ldots , K_r\}$. Each $(G \cup K_i), i=1,\ldots ,r$ is a knot (a set of intersected cycles) of the graph. The arrangement of the clauses of $\overline{T}$ allows the decomposition of $G$ in knots and provides a way of computing $\sharp $2Sat(F) in an incremental way. Our procedure has a bottom-up orientation for the computation of $\sharp $2Sat($F$). It begins with $F_0 = T$. In each iteration of the procedure, a new clause $C_i \in \overline{T}$ is considered in order to form $F_i = (F_{i-1} \wedge C_i)$ and then to compute $\sharp $2Sat$(F_i)$ based on the computation of $\sharp $2Sat$(F_{i-1})$.