Maximizing the Number of Satisfied L-clauses

TL;DR

This paper explores \\L{}-clausal forms in the context of the $k$-SAT problem, proving NP-completeness of minimizing their cost, proposing an instance generator, and empirically analyzing how parameters affect satisfiability and cost.

Contribution

It introduces a new NP-completeness proof for minimizing \\L{}-clausal form costs and presents an instance generation model for experimental analysis.

Findings

Cost decreases exponentially with increased $p$

Generated instances can be satisfiable or unsatisfiable with same ratios

Empirical results reveal parameter effects on satisfiability

Abstract

The $k$-SAT problem for \L{}-clausal forms has been found to be NP-complete if $k\geq 3$. Similar to Boolean CNF formulas, \L{}-clausal forms are important from a theoretical and practical points of view for their expressive power, easy-hard-easy pattern as well as having a phase transition phenomena. In this paper, we investigate further \L{}-clausal forms in terms of instance generation and maximizing the number of satisfied \L{}-clauses. Firstly, we prove that minimizing the cost of \L{}-clausal forms is NP-complete and present an algorithm for the problem. Secondly, we devise an instance generation model to produce \L{}-clausal forms with different values of $k$ and degree of absence of negated terms $\neg(l_1 \oplus \dots \oplus l_m)$ (we call $p$) in each clause. Finally, we conduct empirical investigation to identify the relationship between the cost and other parameters of the instance generator. One of our findings shows that the cost decreases exponentially as $p$ increases, for any clauses to variables ratio. This enables us to generate satisfiable and unsatisfiable instances with the same clauses to variables ratio.