New Approximation Algorithms for Maximum Asymmetric Traveling Salesman and Shortest Superstring

TL;DR

This paper introduces a faster combinatorial approximation algorithm for Max ATSP with a 0.7 guarantee, and derives an improved approximation for the shortest superstring problem using this result.

Contribution

It presents a novel 0.7-approximation algorithm for Max ATSP using innovative techniques like half-edges and edge coloring, and improves the approximation ratio for the shortest superstring problem.

Findings

Achieves a 0.7 approximation for Max ATSP.

Provides a new approximation ratio of approximately 2.434 for SSP.

Introduces techniques of diluting and eliminating subgraphs with half-edges.

Abstract

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 $\frac{7}{10}$-approximation algorithm for Max ATSP. It is based on techniques of {\em eliminating} and {\em diluting} problematic subgraphs with the aid of {\it half-edges} and a method of edge coloring. (A {\it half-edge} of edge $(u,v)$ is informally speaking "either a head or a tail of $(u,v)$".) A novel technique of {\em 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 $\frac 23$, are due to Kaplan, Lewenstein, Shafrir, Sviridenko (2003) and Elbassioni, Paluch, van Zuylen (2012). Using a result by Mucha, which states that an $\alpha$-approximation algorithm for Max ATSP implies a $(2+\frac{11(1-\alpha)}{9-2\alpha})$-approximation algorithm for the shortest superstring problem (SSP), we obtain also a $(2 \frac{33}{76} \approx 2,434)$-approximation algorithm for SSP, beating the previously best known (having an approximation factor equal to $2 \frac{11}{23} \approx 2,4782$.)