Improved Algorithms for the General Exact Satisfiability Problem

TL;DR

This paper introduces faster polynomial and exponential space algorithms for the Generalised Exact Satisfiability (G$i$XSAT) problem, significantly improving the efficiency over previous methods for solving these complex logical problems.

Contribution

The paper presents novel algorithms with improved time complexities for solving G$i$XSAT for i=2,3,4, using polynomial and exponential space, advancing the state of the art.

Findings

Polynomial space algorithms with faster time complexities for G2XSAT, G3XSAT, G4XSAT.

Exponential space algorithms with even faster time complexities for the same problems.

Significant improvement over previous algorithms in terms of runtime efficiency.

Abstract

The Exact Satisfiability problem asks if we can find a satisfying assignment to each clause such that exactly one literal in each clause is assigned $1$, while the rest are all assigned $0$. We can generalise this problem further by defining that a $C^j$ clause is solved iff exactly $j$ of the literals in the clause are $1$ and all others are $0$. We now introduce the family of Generalised Exact Satisfiability problems called G$i$XSAT as the problem to check whether a given instance consisting of $C^j$ clauses with $j \in \{0,1,\ldots,i\}$ for each clause has a satisfying assignment. In this paper, we present faster exact polynomial space algorithms, using a nonstandard measure, to solve G$i$XSAT, for $i\in \{2,3,4\}$, in $O(1.3674^n)$ time, $O(1.5687^n)$ time and $O(1.6545^n)$ time, respectively, using polynomial space, where $n$ is the number of variables. This improves the current state of the art for polynomial space algorithms from $O(1.4203^n)$ time for G$2$XSAT by Zhou, Jiang and Yin and from $O(1.6202^n)$ time for G$3$XSAT by Dahll\"of and from $O(1.6844^n)$ time for G$4$XSAT which was by Dahll\"of as well. In addition, we present faster exact algorithms solving G$2$XSAT, G$3$XSAT and G$4$XSAT in $O(1.3188^n)$ time, $O(1.3407^n)$ time and $O(1.3536^n)$ time respectively at the expense of using exponential space.