Constant Factor Approximation for ATSP with Two Edge Weights

TL;DR

This paper presents a constant factor approximation algorithm for the ATSP with two edge weights, extending previous work to a more general setting and introducing a flow decomposition approach.

Contribution

It introduces a solution to Local-Connectivity ATSP with two edge weights using a flow decomposition theorem, advancing approximation algorithms for ATSP.

Findings

Achieved a constant factor approximation for ATSP with two edge weights.

Developed a flow decomposition theorem for solutions of the Held-Karp relaxation.

Extended the approach for unit weights to two different edge weights.

Abstract

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently by Svensson. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest.