A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs

TL;DR

This paper introduces a near-optimal polynomial-time approximation algorithm for the asymmetric TSP on graphs with bounded genus, improving previous bounds and applicable even without explicit embeddings.

Contribution

It presents the first approximation algorithm for ATSP on graphs with bounded non-orientable genus, achieving better approximation ratios and efficiency without requiring an explicit embedding.

Findings

Achieves an O(log(g)/loglog(g))-approximation for genus-g graphs.

Improves previous approximation bounds for graphs with bounded genus.

Provides an O(1)-approximation algorithm with exponential dependence on genus.

Abstract

We present a near-optimal polynomial-time approximation algorithm for the asymmetric traveling salesman problem for graphs of bounded orientable or non-orientable genus. Our algorithm achieves an approximation factor of O(f(g)) on graphs with genus g, where f(n) is the best approximation factor achievable in polynomial time on arbitrary n-vertex graphs. In particular, the O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation algorithm for genus-g graphs. Our result improves the O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA 2011], which applies only to graphs with orientable genus g; ours is the first approximation algorithm for graphs with bounded non-orientable genus. Moreover, using recent progress on approximating the genus of a graph, our O(log(g) / loglog(g))-approximation can be implemented even without an embedding when the input graph has bounded degree. In contrast, the O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a genus-g embedding as part of the input. Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on graphs of genus g, with running time 2^O(g)*n^O(1).