  We study the problem of finding a tour of $n$ points in which every edge is
long. More precisely, we wish to find a tour that visits every point exactly
once, maximizing the length of the shortest edge in the tour. The problem is
known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997),
motivated by applications in manufacturing and medical imaging. Arkin et al.
gave a $0.5$-approximation for the metric version of the problem and showed
that this is the best possible ratio achievable in polynomial time (assuming $P
\neq NP$). Arkin et al. raised the question of whether a better approximation
ratio can be obtained in the Euclidean plane.
  We answer this question in the affirmative in a more general setting, by
giving a $(1-\epsilon)$-approximation algorithm for $d$-dimensional doubling
metrics, with running time $\tilde{O}\big(n^3 + 2^{O(K \log K)}\big)$, where $K
\leq \left( \frac{13}{\epsilon} \right)^d$. As a corollary we obtain (i) an
efficient polynomial-time approximation scheme (EPTAS) for all constant
dimensions $d$, (ii) a polynomial-time approximation scheme (PTAS) for
dimension $d = \log\log{n}/c$, for a sufficiently large constant $c$, and (iii)
a PTAS for constant $d$ and $\epsilon = \Omega(1/\log\log{n})$. Furthermore, we
show the dependence on $d$ in our approximation scheme to be essentially
optimal, unless Satisfiability can be solved in subexponential time.
