The traveling salesman problem for lines, balls and planes

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.