An ETH-Tight Exact Algorithm for Euclidean TSP

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 $n^{O(n^{1-1\mathrm{/}d})}$ running time was presented for any $d\ge 2$. 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 $2^{O(n^{1-1\mathrm{/}d-ϵ})}$ algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of Euclidean TSP by giving a $2^{O(n^{1-1\mathrm{/}d})}$ algorithm and by showing that a $2^{o(n^{1-1\mathrm{/}d})}$ algorithm does not exist unless ETH fails.