A domination algorithm for 0, 1‐instances of the travelling salesman problem1. Issue 3 (8th October 2015) - Kühn, Daniela; Osthus, Deryk; Patel, Viresh
Abstract: 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 K n on n vertices, our algorithm outputs a Hamilton cycle H * of K n with the following property: the proportion of Hamilton cycles of K n 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 ( − ( log n ) C ) in the result above. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 427–453, 2016