Bibliographic record
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 captures 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-ε} n})-hard to approximate, where n is the number of vertices, under a standard complexity assumption, by Elkin and Peleg (Theory of Computing Systems, 2007). Our results for buy-at-bulk spanners are the following. - When the edge lengths are integral with magnitude polynomial in n we present: 1) An Õ(n^{4/5 + ε})-approximation polynomial-time randomized algorithm for uniform demands. 2) An Õ(k^{1/2 + ε})-approximation polynomial-time randomized algorithm for general demands, where k is the number of terminal pairs. This can be improved to an Õ(k^{ε})-approximation algorithm for the single-source problem. The same approximation ratios hold in the online setting. - When the edge lengths are rational and well-conditioned, we present an Õ(k^{1/2 + ε})-approximation polynomial-time randomized algorithm that may slightly violate the distance constraints. The result can be improved to an Õ(k^ε)-approximation algorithm for the single-source problem. The same approximation ratios hold for the online setting when the condition number is given in advance. To the best of our knowledge, these are the first sublinear factor approximation algorithms for directed buy-at-bulk spanners. We allow the edge lengths to be negative and the demands to be non-unit, unlike the previous literature. Our approximation ratios match the state-of-the-art ratios in special cases, namely, buy-at-bulk network design by Antonakopoulos (WAOA, 2010) and (online) weighted spanners by Grigorescu, Kumar, and Lin (APPROX 2023). Furthermore, we improve the competitive ratio for online buy-at-bulk by Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018) by a factor of log R, where R is the ratio between the maximum demand and the minimum demand.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.017 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".