Directed Buy-at-Bulk Spanners
Elena Grigorescu, Nithish Kumar, Young-San Lin
TL;DR
This work develops a unified framework for directed buy-at-bulk spanners, incorporating distance constraints and the possibility of negative edge lengths. The authors introduce two independent, of-interest problems—Minimum-Density Distance-Constrained Junction Trees and Resource-Constrained Shortest Path with Negative Consumption—and build algorithms that achieve sublinear approximation factors in both the domain [poly(n)] and real-valued budgets. Key results include a randomized tilde-omitted $\tilde{O}(n^{4/5+\varepsilon})$-approximation for unit-demand buy-at-bulk spanners on $[\mathsf{poly}(n)]_{\pm}$ and $\tilde{O}(k^{1/2+\varepsilon})$-type guarantees for real budgets, along with a $\tilde{O}(k^{\varepsilon})$-approximation for single-source variants. The techniques combine layered-graph constructions, height-reduction, LP formulations with rounding, and RCSP-based separation oracles, enabling robust handling of negative edge lengths and complex budget constraints with provable performance guarantees. This advances the practical applicability of buy-at-bulk and spanner concepts to networks with economies of scale, latency considerations, and negative resources, and it opens avenues for further research on multi-resource routing under challenging cost and distance constraints.
Abstract
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.