A Faster Exact Algorithm to Count X3SAT Solutions

TL;DR

This paper introduces a faster exact algorithm for counting solutions to the X3SAT problem, improving the runtime from 1.1487^n to 1.1120^n by leveraging graph bisection width results.

Contribution

The paper presents a novel algorithm for #X3SAT that reduces the exponential base, offering a more efficient solution compared to previous methods.

Findings

The new algorithm runs in O(1.1120^n) time.

It uses graph bisection width results for degree-3 graphs.

Fewer branching cases are needed in the algorithm.

Abstract

The Exact Satisfiability problem, XSAT, is defined as the problem of finding a satisfying assignment to a formula in CNF such that there is exactly one literal in each clause assigned to be 1 and the other literals in the same clause are set to 0. If we restrict the length of each clause to be at most 3 literals, then it is known as the X3SAT problem. In this paper, we consider the problem of counting the number of satisfying assignments to the X3SAT problem, which is also known as #X3SAT. The current state of the art exact algorithm to solve #X3SAT is given by Dahll\"of, Jonsson and Beigel and runs in $O(1.1487^n)$, where $n$ is the number of variables in the formula. In this paper, we propose an exact algorithm for the #X3SAT problem that runs in $O(1.1120^n)$ with very few branching cases to consider, by using a result from Monien and Preis to give us a bisection width for graphs with at most degree 3.