The Unbounded Integrality Gap of a Semidefinite Relaxation of the Traveling Salesman Problem

We study a semidefinite programming relaxation of the traveling salesman problem intro-duced by de Klerk, Pasechnik, and Sotirov [8] and show that their relaxation has an unboundedintegrality gap. In particular, we give a family of instances such that the gap increases linearlywith n. To obtain this result, we search for feasible solutions within a highly structured class ofmatrices; the problem of finding such solutions reduces to finding feasible solutions for a relatedlinear program, which we do analytically. The solutions we find imply the unbounded integral-ity gap. Further, they imply several corollaries that help us better understand the semidefiniteprogram and its relationship to other TSP relaxations. Using the same technique, we show thata more general semidefinite program introduced by de Klerk, de Oliveira Filho, and Pasechnik[7] for the k-cycle cover problem also has an unbounded integrality gap.