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

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

2005-03-27

arxiv.org/abs/cond-mat/0503634v1

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

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.

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