Positive 1-in-3-SAT admits a non-trivial Kernel

TL;DR

This paper demonstrates that positive 1-in-3 SAT problems can be efficiently reduced to smaller equivalent instances using algebraic methods, leading to non-trivial kernels and improved complexity bounds.

Contribution

It introduces a novel algebraic approach to reduce positive 1-in-3 SAT to smaller instances, establishing the existence of non-trivial kernels for this problem.

Findings

Reduced positive 1-in-3 SAT to instances with at most 2/3 variables

Provided complexity bounds for counting solutions

Established non-trivial kernel for 1-in-3 SAT

Abstract

We illustrate the strength of Algebraic Methods, adapting Gaussian Elimination and Substitution to the problem of Exact Boolean Satisfiability. For 1-in-3 SAT with non-negated literals we are able to obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to Equality only. Both Gaussian Elimination and Substitution may be used in a processing step, followed by a type of brute-force approach on the kernel thus obtained. Our method shows that Positive instances of 1-in-3 SAT may be reduced to significantly smaller instances of I.P.E. in the following sense. Any such instance of $|V|$ variables and $|C|$ clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with Equality, of size at most $2/3|V|$ variables and at most $|C|$ clauses. We obtain an upper bound for the complexity of counting, $O(2\kappa r 2^{(1-\kappa) r})$ for number of variables $r$ and clauses to variables ratio $\kappa$. We proceed to define formally the notion of a non-trivial kernel, defining the problems considered as Constraint Satisfaction Problems. We conclude showing the methods presented here, giving a non-trivial kernel for positive 1-in-3 SAT, imply the existence of a non-trivial kernel for 1-in-3 SAT.