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(nlog⁡n)-time algorithms exist for paths and trees. We show that o(n2) queries cannot provide a better than 10/9-approximation for trees when k≥3. For any constant ε>0, we design a linear-time (1+ε)-approximation algorithm for paths when k=o(log⁡n), thus establishing a dichotomy between paths and trees for k≥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(nklog⁡n).