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arXiv · 1107.1628

A Proof of the Boyd-Carr Conjecture

Frans Schalekamp·David P. Williamson·Anke van Zuylen
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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.

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Frans Schalekamp, David P. Williamson, Anke van Zuylen. 2011-07-08. A Proof of the Boyd-Carr Conjecture. https://arxiv.org/abs/1107.1628

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