# Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps

## TL;DR

This paper investigates the effectiveness of semidefinite programming relaxations for the TSP, revealing unbounded integrality gaps on certain instances and establishing limitations of current SDP approaches.

## Contribution

It introduces simplicial TSP instances that demonstrate unbounded integrality gaps for SDP relaxations, highlighting their limitations compared to the subtour LP.

## Key findings

- SDP relaxations can have unbounded integrality gaps on specific TSP instances.
- Simplicial TSP instances serve as a benchmark for testing SDP relaxations.
- Subtour LP performs well on simplicial instances, unlike SDP relaxations.

## Abstract

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, e.g., algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [9] present an SDP based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program, but generally dominates it on small instances. We provide a family of \emph{simplicial TSP instances} that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in Section 2 of Sotirov [24]. In contrast, the subtour LP performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP.