We present a framework that unifies directed buy-at-bulk network design and directed spanner problems, namely, \emph{buy-at-bulk spanners}. The goal is to find a minimum-cost routing solution for network design problems that capture economies at scale, while satisfying demands and distance constraints for terminal pairs. A more restricted version of this problem was shown to be $O(2^{{\log^{1-\ep} n}})$-hard to approximate, where $n$ is the number of vertices, under a standard complexity assumption, due to Elkin and Peleg (Theory of Computing Systems, 2007). To the best of our knowledge, our results are the first sublinear factor approximation algorithms for directed buy-at-bulk spanners. Furthermore, these results hold even when we allow the edge lengths to be \emph{negative}, unlike the previous literature for spanners. Our approximation ratios match the state-of-the-art ratios in special cases, namely, buy-at-bulk network design by Antonakopoulos (WAOA, 2010) and weighted spanners by Grigorescu, Kumar, and Lin (APPROX 2023). Our results are based on new approximation algorithms for the following two problems that are of independent interest: \emph{minimum-density distance-constrained junction trees} and \emph{resource-constrained shortest path with negative consumption}.

翻译：我们提出了一个统一有向批量购买网络设计与有向生成子图问题的框架，即“批量购买生成子图”。其目标是在满足终端对的需求和距离约束的同时，为体现规模经济效应的网络设计问题寻找最小成本路由方案。在标准复杂度假设下，Elkin 和 Peleg (Theory of Computing Systems, 2007) 曾证明该问题的一个更受限版本是 $O(2^{{\log^{1-\ep} n}})$-难近似的，其中 $n$ 为顶点数。据我们所知，我们的结果是有向批量购买生成子图问题的首个次线性因子近似算法。此外，与以往生成子图文献不同，即便允许边长为负值，这些结果依然成立。我们的近似比与以下特例中的最优结果相匹配：Antonakopoulos (WAOA, 2010) 研究的批量购买网络设计问题和 Grigorescu、Kumar 与 Lin (APPROX 2023) 研究的加权生成子图问题。我们的结果基于以下两个具有独立意义的问题的新近似算法：“最小密度距离约束联结树”和“负消耗资源约束最短路径”。