An ETH-Tight Exact Algorithm for Euclidean TSP

TL;DR

This paper presents a new exact algorithm for Euclidean TSP that is optimal up to constants, matching the known lower bounds and settling the problem's complexity status.

Contribution

The authors introduce an ETH-tight exact algorithm for Euclidean TSP in any dimension, establishing both upper and lower bounds.

Findings

Algorithm runs in $2^{O(n^{1-1/d})}$ time.

No faster algorithm exists unless ETH fails.

Settles the complexity of Euclidean TSP up to constants.

Abstract

We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was presented for any $d\geq 2$. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Euclidean TSP, except for a lower bound stating that the problem admits no $2^{O(n^{1-1/d-\epsilon})}$ algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of Euclidean TSP by giving a $2^{O(n^{1-1/d})}$ algorithm and by showing that a $2^{o(n^{1-1/d})}$ algorithm does not exist unless ETH fails.