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 exp(−(logn)C) in the result above.Comment: 29 pages (final version to appear in Random Structures and
Algorithms