In the \emph{$k$-Diameter-Optimally Augmenting Tree Problem} we are given a tree $T$ of $n$ vertices as input. The tree is embedded in an unknown \emph{metric} space and we have unlimited access to an oracle that, given two distinct vertices $u$ and $v$ of $T$, can answer queries reporting the cost of the edge $(u,v)$ in constant time. We want to augment $T$ with $k$ shortcuts in order to minimize the diameter of the resulting graph. For $k=1$, $O(n \log n)$ time algorithms are known both for paths [Wang, CG 2018] and trees [Bil\`o, TCS 2022]. In this paper we investigate the case of multiple shortcuts. We show that no algorithm that performs $o(n^2)$ queries can provide a better than $10/9$-approximate solution for trees for $k\geq 3$. For any constant $\varepsilon > 0$, we instead design a linear-time $(1+\varepsilon)$-approximation algorithm for paths and $k = o(\sqrt{\log n})$, thus establishing a dichotomy between paths and trees for $k\geq 3$. We achieve the claimed running time by designing an ad-hoc data structure, which also serves as a key component to provide a linear-time $4$-approximation algorithm for trees, and to compute the diameter of graphs with $n + k - 1$ edges in time $O(n k \log n)$ even for non-metric graphs. Our data structure and the latter result are of independent interest.

翻译：在\textbf{$k$-直径最优增强树问题}中，给定一棵包含$n$个顶点的树$T$作为输入。该树嵌入于一个未知的\textbf{度量}空间，且我们可无限次访问一个预言机，该预言机对任意两个不同顶点$u$和$v$，能在常数时间内查询并返回边$(u,v)$的代价。我们的目标是用$k$条捷径增强树$T$，以最小化所得图的直径。对于$k=1$，已有$O(n \log n)$时间算法分别适用于路径[Wang, CG 2018]和树[Bil\`o, TCS 2022]。本文研究多条捷径的情形。我们证明：当$k\geq 3$时，执行$o(n^2)$次查询的算法无法为树提供优于$10/9$近似比的解。对于任意常数$\varepsilon > 0$，我们为路径且$k = o(\sqrt{\log n})$设计了线性时间的$(1+\varepsilon)$近似算法，从而揭示了$k\geq 3$时路径与树之间的二分性。通过设计一种特定数据机构，我们实现了所述运行时间；该数据机构同时作为关键组件，用于提供树的线性时间$4$近似算法，并可在$O(n k \log n)$时间内计算含有$n+k-1$条边的非度量图的直径。我们的数据机构及后一结果具有独立研究价值。