The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

TL;DR

This paper presents a randomized polynomial-time approximation scheme for the Traveling Salesman Problem in metric spaces with bounded intrinsic dimension, extending known results from Euclidean spaces to more general low-dimensional metric spaces.

Contribution

It introduces a new algorithm showing that TSP is efficiently approximable in low-dimensional metric spaces, regardless of the specific geometry.

Findings

The algorithm achieves a (1+eps)-approximation in polynomial time for fixed eps.

The result generalizes Euclidean TSP algorithms to arbitrary low-dimensional metric spaces.

It resolves an open problem about the complexity of metric TSP in low-dimensional spaces.

Abstract

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04).