# Characterizing the Integrality Gap of the Subtour LP for the Circulant Traveling Salesman Problem

## TL;DR

This paper proves the conjecture that the integrality gap of the subtour LP for circulant TSP instances is exactly 2, providing explicit solutions and revealing that degree constraints do not improve the relaxation.

## Contribution

It explicitly solves the subtour LP for circulant TSP instances, confirming the integrality gap as 2 and showing degree constraints are ineffective in tightening the relaxation.

## Key findings

- Integrality gap of subtour LP on circulant TSP is exactly 2.
- Explicit optimal solution to the subtour LP is constructed.
- Degree constraints do not strengthen the LP on circulant instances.

## Abstract

We consider the integrality gap of the subtour LP relaxation of the Traveling Salesman Problem restricted to circulant instances. De Klerk and Dobre conjectured that the value of the optimal solution to the subtour LP on these instances is equal to an entirely combinatorial lower bound from Van der Veen, Van Dal, and Sierksma. We prove this conjecture by giving an explicit optimal solution to the subtour LP. We then use it to show that the integrality gap of the subtour LP is 2 on circulant instances, making such instances one of the few non-trivial classes of TSP instances for which the integrality gap of the subtour LP is exactly known. We also show that the degree constraints do not strengthen the subtour LP on circulant instances, mimicking the parsimonious property of metric, symmetric TSP instances shown in Goemans and Bertsimas in a distinctly non-metric set of instances.