We present a framework that unifies directed buy-at-bulk network design and directed spanner problems, namely, 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 $O(2^{{\log^{1-\varepsilon} n}})$-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 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: minimum-density distance-constrained junction trees and resource-constrained shortest path with negative consumption.

翻译：我们提出一个统一有向批量购买网络设计问题与有向支撑子图问题的框架，即批量购买支撑子图。其目标是在满足终端对的需求与距离约束的同时，为体现规模经济效应的网络设计问题寻求最小成本路由方案。该问题的一个更受限版本被证明在标准复杂性假设下是 $O(2^{{\log^{1-\varepsilon} n}})$ 难近似的（Elkin 与 Peleg, Theory of Computing Systems, 2007）。据我们所知，我们的结果是针对有向批量购买支撑子图的首个次线性因子近似算法。此外，与以往支撑子图相关文献不同，我们的结果在允许边长为负的情况下依然成立。我们的近似比与特殊情况下的最优结果相匹配，即 Antonakopoulos (WAOA, 2010) 的批量购买网络设计问题以及 Grigorescu, Kumar, 与 Lin (APPROX 2023) 的加权支撑子图问题。我们的结果基于以下两个具有独立价值的问题的新近似算法：最小密度距离约束联结树与负消耗资源约束最短路径。