An improved approximation algorithm for ATSP

TL;DR

This paper improves the approximation ratio for the asymmetric traveling salesman problem from 506 to nearly 22, simplifying the algorithm and reducing the integrality gap.

Contribution

It introduces a simplified reduction to vertebrate pairs and a refined algorithm, significantly enhancing the approximation ratio for ATSP.

Findings

Approximation ratio improved from 506 to 22+ε.

Simpler reduction to vertebrate pairs.

Lowered integrality ratio upper bound from 319 to 22.

Abstract

We revisit the constant-factor approximation algorithm for the asymmetric traveling salesman problem by Svensson, Tarnawski, and V\'egh. We improve on each part of this algorithm. We avoid the reduction to irreducible instances and thus obtain a simpler and much better reduction to vertebrate pairs. We also show that a slight variant of their algorithm for vertebrate pairs has a much smaller approximation ratio. Overall we improve the approximation ratio from $506$ to $22+\epsilon$ for any $\epsilon > 0$. This also improves the upper bound on the integrality ratio from $319$ to $22$.