The traveling salesman problem for lines, balls and planes

TL;DR

This paper introduces new approximation algorithms for the traveling salesman problem with neighborhoods, including hyperplanes, lines, and balls in higher dimensions, providing first known approximations and improved bounds.

Contribution

The paper presents the first approximation algorithms for TSPN with hyperplanes, lines, and balls in higher dimensions, and improves existing bounds for unit disks in the plane.

Findings

Constant-factor approximation for hyperplanes in fixed dimensions

Polylogarithmic approximation for lines in any dimension

Constant-factor approximation for unit balls in fixed dimensions

Abstract

We revisit the traveling salesman problem with neighborhoods (TSPN) and propose several new approximation algorithms. These constitute either first approximations (for hyperplanes, lines, and balls in $\mathbb{R}^d$, for $d\geq 3$) or improvements over previous approximations achievable in comparable times (for unit disks in the plane). \smallskip (I) Given a set of $n$ hyperplanes in $\mathbb{R}^d$, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in $O(n)$ time, when $d$ is constant. \smallskip (II) Given a set of $n$ lines in $\mathbb{R}^d$, a TSP tour whose length is at most $O(\log^3 n)$ times the optimal can be computed in polynomial time for all $d$. \smallskip (III) Given a set of $n$ unit balls in $\mathbb{R}^d$, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in polynomial time, when $d$ is constant.