On semidefinite programming relaxations of the traveling salesman problem

TL;DR

This paper introduces a new semidefinite programming relaxation for the symmetric traveling salesman problem that outperforms previous SDP relaxations and is not dominated by the Held-Karp LP bound, offering a novel approach to TSP bounds.

Contribution

A new SDP relaxation for TSP derived from a relaxation of the quadratic assignment problem, which dominates existing SDP bounds and is incomparable with the Held-Karp LP bound.

Findings

The new SDP relaxation dominates previous SDP bounds.

The new SDP bound is not dominated by the Held-Karp LP bound.

The relaxation is obtained via a relaxation of the quadratic assignment problem.

Abstract

We consider a new semidefinite programming (SDP) relaxation of the symmetric traveling salesman problem (TSP) that may be obtained via an SDP relaxation of the more general quadratic assignment problem (QAP). We show that the new relaxation dominates the one in [D. Cvetkovic, M. Cangalovic, and V. Kovacevic-Vujcic, Semidefinite programming methods for the symmetric traveling salesman problem, in Proc. 7th Int. IPCO Conference, Springer, London, 1999, pp. 126--136]. Unlike the bound of Cvetkovic et al., the new SDP bound is not dominated by the Held-Karp linear programming bound, or vice versa.