A Linear Complementarity Theorem to solve any Satisfiability Problem in conjunctive normal form in polynomial time

TL;DR

This paper presents a polynomial-time method for solving any conjunctive normal form satisfiability problem by reducing it to a linear complementarity problem and solving it via linear programming, under certain conditions.

Contribution

It introduces a proof that these conditions are always satisfied for this problem class, enabling polynomial-time solutions.

Findings

Satisfiability problems can be reduced to LCP and solved as LP.

Necessary and sufficient conditions for solution validity are established.

The approach confirms polynomial-time solvability for all instances in the class.

Abstract

Any satisfiability problem in conjunctive normal form can be solved in polynomial time by reducing it to a 3-sat formulation and transforming this to a Linear Complementarity problem (LCP) which is then solved as a linear program (LP). Any instance in this problem class, reduced to a LCP may be solved, provided certain necessary and sufficient conditions hold. The proof that these conditions will be satisfied for all problems in this class is the contribution of this paper and this derivation requires a nonlinear Instrumentalist methodology rather than a Realistic one and confirms the advantages of a Variational Inequalities implementation.