A Proof of the Boyd-Carr Conjecture
Data Structures and Algorithms
2011-07-11 v1
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.
Cite
@article{arxiv.1107.1628,
title = {A Proof of the Boyd-Carr Conjecture},
author = {Frans Schalekamp and David P. Williamson and Anke van Zuylen},
journal= {arXiv preprint arXiv:1107.1628},
year = {2011}
}