On the Integrality Gap of the Subtour LP for the 1,2-TSP

TL;DR

This paper investigates the integrality gap of the subtour LP relaxation for the 1,2-TSP, providing new bounds and conjectures that improve understanding of this special case and its relation to the general TSP.

Contribution

The paper proposes a new conjecture that the integrality gap is 10/9 for the 1,2-TSP and establishes bounds of 7/6 and 5/4 under certain structural assumptions.

Findings

Conjecture that the integrality gap is 10/9 for 1,2-TSP.

Proven bounds of 7/6 and 5/4 on the integrality gap under specific conditions.

Computational evidence suggests the gap is at most 10/9 for instances with up to 12 cities.

Abstract

In this paper, we study the integrality gap of the subtour LP relaxation for the traveling salesman problem in the special case when all edge costs are either 1 or 2. For the general case of symmetric costs that obey triangle inequality, a famous conjecture is that the integrality gap is 4/3. Little progress towards resolving this conjecture has been made in thirty years. We conjecture that when all edge costs $c_{ij}\in \{1,2\}$, the integrality gap is $10/9$. We show that this conjecture is true when the optimal subtour LP solution has a certain structure. Under a weaker assumption, which is an analog of a recent conjecture by Schalekamp, Williamson and van Zuylen, we show that the integrality gap is at most $7/6$. When we do not make any assumptions on the structure of the optimal subtour LP solution, we can show that integrality gap is at most $5/4$; this is the first bound on the integrality gap of the subtour LP strictly less than $4/3$ known for an interesting special case of the TSP. We show computationally that the integrality gap is at most $10/9$ for all instances with at most 12 cities.