A domination algorithm for {0,1}-instances of the travelling salesman problem

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We present an approximation algorithm for {0,1}-instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial-time algorithm which has domination ratio 1-n⁻¹/²⁹. In other words, given a {0,1}-edge-weighting of the complete graph Kₙ on n vertices, our algorithm outputs a Hamilton cycle H* of Kₙ with the following property: the proportion of Hamilton cycles of Kₙ whose weight is smaller than that of H* is at most n⁻¹/²⁹. Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial-time algorithm with domination ratio 1/2-o(1) for arbitrary edge-weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant C such that n⁻¹/²⁹ cannot be replaced by (-(n)ᶜ) in the result above.