Extending Exact Integrality Gap Computations for the Metric TSP

TL;DR

This paper extends the verification of the integrality gap conjecture for the metric TSP's subtour relaxation to larger instances, providing new extreme point enumerations and supporting the 4/3-Conjecture.

Contribution

It advances the enumeration of extreme points of the subtour polytope for larger vertex counts, confirming and correcting previous results, and supports the 4/3-Conjecture.

Findings

Confirmed results up to n=10 vertices.

Identified missing extreme points for n=11 and n=12.

Extended enumeration to instances with up to 14 vertices, and up to 17 for half-integral vertices.

Abstract

The subtour relaxation of the traveling salesman problem (TSP) plays a central role in approximation algorithms and polyhedral studies of the TSP. A long-standing conjecture asserts that the integrality gap of the subtour relaxation for the metric TSP is exactly 4/3. In this paper, we extend the exact verification of this conjecture for small numbers of vertices. Using the framework introduced by Benoit and Boyd in 2008, we confirm their results up to n=10. We further show that for n=11 and n=12, the published lists of extreme points of the subtour polytope are incomplete: one extreme point is missing for n=11 and twenty-two extreme points are missing for n=12. We extend the enumeration of the extreme points of the subtour polytope to instances with up to 14 vertices in the general case. Restricted to half-integral vertices, we extend the enumeration of extreme points up to n=17. Our results provide additional support for the 4/3-Conjecture. Our lists of extreme points are available on the public bonndata repository (https://doi.org/10.60507/FK2/JK95PC).