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

TL;DR

This paper demonstrates that a specific semidefinite programming relaxation for the Traveling Salesman Problem has an unbounded integrality gap, revealing limitations of this approach and providing insights into its relationship with other relaxations.

Contribution

The authors prove that the semidefinite relaxation by de Klerk, Pasechnik, and Sotirov has an unbounded integrality gap, using a novel analytical approach to construct feasible solutions.

Findings

The integrality gap increases linearly with problem size n.

The relaxation's limitations are characterized through structured feasible solutions.

Similar unbounded gap results are shown for related semidefinite programs.

Abstract

We study a semidefinite programming relaxation of the traveling salesman problem introduced by de Klerk, Pasechnik, and Sotirov [8] and show that their relaxation has an unbounded integrality gap. In particular, we give a family of instances such that the gap increases linearly with $n$. To obtain this result, we search for feasible solutions within a highly structured class of matrices; the problem of finding such solutions reduces to finding feasible solutions for a related linear program, which we do analytically. The solutions we find imply the unbounded integrality gap. Further, they imply several corollaries that help us better understand the semidefinite program and its relationship to other TSP relaxations. Using the same technique, we show that a 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.