Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget $k$, which $k$ edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in $k$. Our main result is a $(2 \sqrt[r]{2} \ k^{1/r},2r)$-bicriteria approximation that runs in $O(n^3 \log n)$ time, for all $r \geq 1$. In other words, if $t^*$ is the minimum dilation after adding any $k$ edges to a graph, then our algorithm adds $O(k^{1+1/r})$ edges to the graph to obtain a dilation of $2rt^*$. Moreover, our analysis of the algorithm is tight under the Erd\H{o}s girth conjecture.

翻译：生成图构造主要关注网络的初始设计。然而，网络往往会随时间推移而改进。本文聚焦于网络改进阶段：给定一个图与预算$k$，我们应添加哪$k$条边才能使图的扩张最小化？Gudmundsson与Wong [TALG'22] 针对该问题给出了首个积极结果，但其近似因子与$k$呈线性关系。我们的主要成果是提出一个$(2 \sqrt[r]{2} \ k^{1/r},2r)$-双准则近似算法，该算法对所有$r \geq 1$在$O(n^3 \log n)$时间内运行。换言之，若$t^*$是在图中添加任意$k$条边后获得的最小扩张值，则我们的算法通过添加$O(k^{1+1/r})$条边使图获得$2rt^*$的扩张值。此外，根据Erd\H{o}s围长猜想，我们的算法分析具有紧致性。