Given a weighted graph $G$, a minimum weight $\alpha$-spanner is a least-weight subgraph $H\subseteq G$ that preserves minimum distances between all node pairs up to a factor of $\alpha$. There are many results on heuristics and approximation algorithms, including a recent investigation of their practical performance [20]. Exact approaches, in contrast, have long been denounced as impractical: The first exact ILP (integer linear program) method [48] from 2004 is based on a model with exponentially many path variables, solved via column generation. A second approach [2], modeling via arc-based multicommodity flow, was presented in 2019. In both cases, only graphs with 40-100 nodes were reported to be solvable. In this paper, we briefly report on a theoretical comparison between these two models from a polyhedral point of view, and then concentrate on improvements and engineering aspects. We evaluate their performance in a large-scale empirical study. We report that our tuned column generation approach, based on multicriteria shortest path computations, is able to solve instances with over 16000 nodes within 13 minutes. Furthermore, now knowing optimal solutions for larger graphs, we are able to investigate the quality of the strongest known heuristic on reasonably sized instances for the first time.

翻译：给定一个加权图$G$，最小权重$\alpha$-生成子图是指满足$H\subseteq G$且保持所有节点对之间最短距离至多扩大$\alpha$倍的最小权重子图。现有研究已提出了多种启发式算法与近似算法，并对其实际性能进行了近期探讨[20]。相比之下，精确求解方法长期被认为缺乏实用性：2004年提出的首个精确整数线性规划方法[48]基于具有指数级路径变量的模型，通过列生成求解；2019年提出的第二种方法[2]则采用基于弧的多商品流建模。这两种方法此前仅能求解40-100个节点的图。本文首先从多面体理论视角简要比较这两种模型，继而重点阐述算法改进与工程实现细节。我们通过大规模实验评估了算法性能，结果表明：基于多准则最短路径计算的优化列生成方法能在13分钟内求解超过16000个节点的实例。此外，通过首次获得较大规模图的最优解，我们得以在合理规模实例上考察当前最强启发式算法的求解质量。