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

Henri Orland (1) , Michel Bauer (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

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 problems.

Domaines: Mécanique statistique [cond-mat.stat-mech]; Systèmes désordonnés et réseaux de neurones [cond-mat.dis-nn]

Henri Orland, Michel Bauer. Checking for optimal solutions in some $NP$-complete problems. Physical Review Letters, 2005, 95 (10), pp.107202. ⟨10.1103/PhysRevLett.95.107202⟩. ⟨hal-00004593⟩