A Proof of the Boyd-Carr Conjecture

TL;DR

This paper proves the Boyd-Carr conjecture, establishing that the worst-case ratio of the optimal 2-matching to the subtour LP relaxation in the traveling salesman problem is at most 10/9, advancing understanding of integrality gaps.

Contribution

We prove the Boyd-Carr conjecture, providing the first tight bound on the ratio between optimal 2-matching and the subtour LP for symmetric TSP instances.

Findings

The ratio of optimal 2-matching to the subtour LP is at most 10/9.

In cases with no cut edge, the ratio is also at most 10/9.

This result confirms the conjectured upper bound on the integrality gap.

Abstract

Determining the precise integrality gap for the subtour LP relaxation of the traveling salesman problem is a significant open question, with little progress made in thirty years in the general case of symmetric costs that obey triangle inequality. Boyd and Carr [3] observe that we do not even know the worst-case upper bound on the ratio of the optimal 2-matching to the subtour LP; they conjecture the ratio is at most 10/9. In this paper, we prove the Boyd-Carr conjecture. In the case that a fractional 2-matching has no cut edge, we can further prove that an optimal 2-matching is at most 10/9 times the cost of the fractional 2-matching.