Faster Approximation Scheme for Euclidean $k$-TSP

TL;DR

This paper presents a faster approximation scheme for the Euclidean $k$-TSP problem, improving previous algorithms with a more efficient runtime that is also tight under Gap-ETH assumptions.

Contribution

It introduces a new approximation scheme for Euclidean $k$-TSP with significantly improved running time over prior work, and discusses its derandomization.

Findings

Achieves a runtime of $n imes 2^{O(1/\varepsilon^{d-1})} imes ( ext{log } n)^{2d^2 imes 2^d}$.

Improves upon Arora's 1998 approximation scheme.

Algorithm is Gap-ETH tight and can be derandomized with increased runtime.

Abstract

In the Euclidean $k$-traveling salesman problem ($k$-TSP), we are given $n$ points in the $d$-dimensional Euclidean space, for some fixed constant $d\geq 2$, and a positive integer $k$. The goal is to find a shortest tour visiting at least $k$ points. We give an approximation scheme for the Euclidean $k$-TSP in time $n\cdot 2^{O(1/\varepsilon^{d-1})} \cdot(\log n)^{2d^2\cdot 2^d}$. This improves Arora's approximation scheme of running time $n\cdot k\cdot (\log n)^{\left(O\left(\sqrt{d}/\varepsilon\right)\right)^{d-1}}$ [J. ACM 1998]. Our algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor $O(n^d)$.