A Parametric Polynomial Deterministic Algorithm for #2SAT

Abstract

Counting models for two Conjunctive Normal Form formulae (2-CFs), known as the #2SAT problem, is a classic #P complete problem. It is known that if the constraint graph of a 2-CF F is acyclic or contains loops and parallel edges, $\#2SAT(F)$ can be computed efficiently. In this paper we address the cyclic case different from loops and parallel edges.

If the constraint graph G of a 2-CF F is cyclic, T a spanning tree plus loops and parallel edges of G and $\overline{T}=G\setminus T$, what we called its cotree, we show that by building a set partition $\cup T_i$ of $\overline{T}$, where each $T_i$ of the partition is formed by the frond edges of the cycles that are chained via other intersected cycles, then a parametric polynomial deterministic procedure for computing #2SAT with time complexity for the worst case of $O(2^{k} \cdot poly(|E(T)|))$ can be obtained, where poly is a polynomial function, and k is the cardinality of the largest set in the partition.

This method shows that #2SAT is in the class of fixed-parameter tratable (FPT) problems, where the fixed-parameter k in our proposal, depends on the number of edges of a subcotree of a decomposition of the constraint graph (tree+loops+parallel:cotree) associated to the formula.