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(n(2)) queries cannot provide a better than 10/9-approximation for trees when k >= 3. For any constant epsilon > 0, we design a linear-time (1 + epsilon)-approximation algorithm for paths when k = o(root logn), 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 4approximation algorithm for trees, and to compute the diameter of (possibly non-metric) graphs with n + k - 1 edges in time O (nk log n). (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).