A Linear Time Gap-ETH-Tight Approximation Scheme for Euclidean TSP

TL;DR

This paper presents a new randomized approximation scheme for Euclidean TSP that runs in linear time with respect to the number of points, matching the best possible dependence on the approximation parameter.

Contribution

It introduces a linear time approximation scheme for Euclidean TSP that achieves the optimal dependence on the approximation parameter, resolving an open problem.

Findings

Achieves a randomized $2^{O(1/\varepsilon^{d-1})} n$ time complexity.

Matches the Gap-ETH tight bound on the dependence of $\varepsilon$.

Provides the first linear time scheme with optimal $\varepsilon$ dependence.

Abstract

The Traveling Salesman Problem (TSP) in the $d$-dimensional Euclidean space is among the oldest and most famous NP-hard optimization problems. In breakthrough works, Arora [J. ACM 1998] and Mitchell [SICOMP 1999] gave the first polynomial time approximation schemes. To improve the running time, Rao and Smith [STOC 1998] gave a randomized $(1/\varepsilon)^{O(1/\varepsilon^{d-1})}\cdot n\log n$ time approximation scheme. Bartal and Gottlieb [FOCS 2013] gave a randomized approximation scheme in $2^{(1/\varepsilon)^{O(d)}} n$ time, which is linear in $n$. Recently, Kisfaludi-Bak, Nederlof, and W\k{e}grzycki [FOCS 2021] gave a randomized approximation scheme in $2^{O(1/\varepsilon^{d-1})} n \log n$ time, achieving a Gap-ETH tight dependence on $\varepsilon$. It is raised as a challenging open question by Kisfaludi-Bak, Nederlof, and W\k{e}grzycki [FOCS 2021] whether a running time of $2^{O(1/\varepsilon^{d-1})}n$ is achievable. We answer their question positively by giving a randomized $2^{O(1/\varepsilon^{d-1})} n$ time approximation scheme for Euclidean TSP.