Finding diameter-reducing shortcuts in trees

In the k-Diameter-Optimally Augmenting Tree Problem we are given a tree T of n vertices embedded in an unknown metric space. An oracle can report the cost of any edge in constant time, and we want to augment T with k shortcuts to minimize the resulting diameter. When $k=1$, $O(n\log n)$-time algorithms exist for paths and trees. We show that $o\left(n^2\right)$ queries cannot provide a better than 10/9-approximation for trees when $k\ge 3$. For any constant $\epsilon >0$, we design a linear-time $(1+\epsilon )$-approximation algorithm for paths when $k=o\left(\sqrt{\log n}\right)$, thus establishing a dichotomy between paths and trees for $k\ge 3$. Our algorithm employs an ad-hoc data structure, which we also use in a linear-time 4-approximation algorithm for trees, and to compute the diameter of (possibly non-metric) graphs with $n+k-1$ edges in time $O(nk\log n)$.