New Approximation Algorithms for Maximum Asymmetric Traveling Salesman and Shortest Superstring

05/21/2020 ∙ by Katarzyna Paluch, et al. ∙ 0

In the maximum asymmetric traveling salesman problem (Max ATSP) we are given a complete directed graph with nonnegative weights on the edges and we wish to compute a traveling salesman tour of maximum weight. In this paper we give a fast combinatorial 7/10-approximation algorithm for Max ATSP. It is based on techniques of eliminating and diluting problematic subgraphs with the aid of half-edges and a method of edge coloring. (A half-edge of edge (u,v) is informally speaking "either a head or a tail of (u,v)".) A novel technique of diluting a problematic subgraph S consists in a seeming reduction of its weight, which allows its better handling. The current best approximation algorithms for Max ATSP, achieving the approximation guarantee of 2/3, are due to Kaplan, Lewenstein, Shafrir, Sviridenko (2003) and Elbassioni, Paluch, van Zuylen (2012). Using a result by Mucha, which states that an α-approximation algorithm for Max ATSP implies a (2+11(1-α)/9-2α)-approximation algorithm for the shortest superstring problem (SSP), we obtain also a (2 33/76≈ 2,434)-approximation algorithm for SSP, beating the previously best known (having an approximation factor equal to 2 11/23≈ 2,4782.)

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