Extending Exact Integrality Gap Computations for the Metric TSP

2026-03-13

AI Summary

This study investigates the long-standing conjecture that the integrality gap of the subtour relaxation for the metric Traveling Salesman Problem (TSP) is 4/3. Building upon the framework of Benoit and Boyd (2008), we systematically analyze the vertex structure of the subtour polytope by integrating polyhedral enumeration, linear programming, and symmetry-based pruning techniques. We correct the known vertex lists for instances with $n = 11$ and $n = 12$, and extend the enumeration to general instances up to $n = 14$ and half-integral instances up to $n = 17$, providing the first complete verification of all vertices in these ranges. Our results confirm that the 4/3 integrality gap conjecture holds for all general instances with $n \leq 14$ and all half-integral instances with $n \leq 17$, offering the strongest empirical support to date for this conjecture.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

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.

Problem

• Traveling Salesman Problem
• Integrality Gap
• Subtour Relaxation
• Metric TSP
• 4/3-Conjecture

Innovation

• integrality gap
• subtour relaxation
• extreme points
• metric TSP
• polyhedral combinatorics