A Fast Exponential Time Algorithm for Max Hamming Distance X3SAT

TL;DR

This paper introduces a significantly faster exponential time algorithm for the Max Hamming Distance problem in X3SAT, improving previous algorithms and efficiently counting solution pairs at each Hamming distance.

Contribution

It presents a new $O(1.3298^n)$ algorithm for Max Hamming Distance X3SAT, advancing the computational efficiency over prior methods.

Findings

Achieved a new exponential time complexity of $O(1.3298^n)$

Algorithm counts pairs of solutions at each Hamming distance

Improves upon previous algorithms with higher runtimes

Abstract

X3SAT is the problem of whether one can satisfy a given set of clauses with up to three literals such that in every clause, exactly one literal is true and the others are false. A related question is to determine the maximal Hamming distance between two solutions of the instance. Dahll\"of provided an algorithm for Maximum Hamming Distance XSAT, which is more complicated than the same problem for X3SAT, with a runtime of $O(1.8348^n)$; Fu, Zhou and Yin considered Maximum Hamming Distance for X3SAT and found for this problem an algorithm with runtime $O(1.6760^n)$. In this paper, we propose an algorithm in $O(1.3298^n)$ time to solve the Max Hamming Distance X3SAT problem; the algorithm actually counts for each $k$ the number of pairs of solutions which have Hamming Distance $k$.