Checking for optimal solutions in some $NP$-complete problems

TL;DR

This paper introduces a polynomial-time algorithm to verify the optimality of solutions for certain NP-complete problems, including the weighted tripartite matching problem and variants of the multiple traveling salesmen problem, using mean-field equations.

Contribution

It presents a novel method leveraging mean-field equations to efficiently check solution optimality in specific NP-complete problems, which was previously non-trivial.

Findings

The mean-field equations become exact at zero temperature.

The proposed algorithm can verify optimality in polynomial time.

Generalization to variants of the multiple traveling salesmen problem is demonstrated.

Abstract

For some weighted $NP$-complete problems, checking whether a proposed solution is optimal is a non-trivial task. Such is the case for the celebrated traveling salesman problem, or the spin-glass problem in 3 dimensions. In this letter, we consider the weighted tripartite matching problem, a well known $NP$-complete problem. We write mean-field finite temperature equations for this model, and show that they become exact at zero temperature. As a consequence, given a possible solution, we propose an algorithm which allows to check in a polynomial time if the solution is indeed optimal. This algorithm is generalized to a class of variants of the multiple traveling salesmen problem.