A Constant-Factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem

TL;DR

This paper presents a constant-factor approximation algorithm for the asymmetric traveling salesman problem (ATSP), confirming the conjectured integrality gap of the standard LP relaxation through a novel reduction approach.

Contribution

It introduces a reduction from ATSP to Subtour Partition Cover, enabling a constant-factor approximation and confirming the LP relaxation's integrality gap conjecture.

Findings

Achieved a constant-factor approximation for ATSP.

Established the LP relaxation's integrality gap as a constant.

Reduced ATSP to a more manageable Subtour Partition Cover problem.

Abstract

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem (ATSP). Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. The main idea of our approach is a reduction to Subtour Partition Cover, an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. We first show that any algorithm for Subtour Partition Cover can be turned into an algorithm for ATSP while only losing a small constant factor in the performance guarantee. Next, we present a reduction from general ATSP instances to structured instances, on which we then solve Subtour Partition Cover, yielding our constant-factor approximation algorithm for ATSP.