The Integrality Gap of the Traveling Salesman Problem is $4/3$ if the LP Solution Has at Most $n+6$ Non-zero Components

TL;DR

This paper proves that the integrality gap of the symmetric metric TSP's LP relaxation is 4/3 when the optimal solution has at most n+6 non-zero components, using a new combined theoretical and computational approach.

Contribution

It establishes the 4/3 integrality gap for instances with limited non-zero components, advancing understanding of the TSP LP relaxation.

Findings

Proves the 4/3 integrality gap under specific sparsity conditions.

Develops a novel methodology combining theory and computation.

Provides insights into the structure of near-optimal solutions.

Abstract

We address the classical Dantzig - Fulkerson - Johnson formulation of the symmetric metric Traveling Salesman Problem and study the integrality gap of its linear relaxation, namely the Subtour Elimination Problem (SEP). This integrality gap is conjectured to be 4/3. We prove that, when solving a problem on n nodes, if the optimal SEP solution has at most n + 6 non-zero components, then the conjecture is true. To establish this result, we devise a new methodology that combines theoretical analysis and computational verification.