Improved Algorithms for the General Exact Satisfiability Problem

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.