Simple cost sharing schemes for multicommodity rent-or-buy and stochastic Steiner tree
Bibliographic record
Abstract
In the multi-commodity rent-or-buy network design problem (MRoB) we are given a network together with a set of k terminal pairs R = (s_1, t_1), ..., (s_k, t_k). The goal is to install capacities on the edges of the network so that a prescribed amount of flow fi can be routed between all terminal pairs si and ti simultaneously. We can either rent capacity on an edge at some cost per unit flow or buy infinite capacity on an edge at some larger fixed cost. The overall objective is to install capacities at a minimum total cost.The version of the stochastic Steiner tree problem (SST) considered here is the Steiner tree problem in the model of two-stage stochastic optimization with recourse. In stage one, there is a known probability distribution on subsets of vertices and we can choose to buy a subset of edges at a given cost. In stage two, a subset of vertices T from the prior known distribution is realized, and additional edges can be bought at a possibly higher cost. The objective is to buy a set of edges in stages one and two so that all vertices in T are connected, and the expected cost is minimized.Gupta et al. (FOCS '03) give a randomized scheme for the MRoB problem that was both used subsequently to improve the approximation ratio for this problem, and extended to yield the best approximation algorithm for SST. One building block of this scheme is a good approximation algorithm for Steiner forests.We present a surprisingly simple 5-approximation algorithm for MRoB and 6-approximation for SST, improving on the best previous guarantees of 6.828 and 12.6, and show that no approximation ratio better than 4.67 can be achieved using the above mentioned randomized scheme in combination with the currently best known Steiner forest approximation algorithms. A key component of our approach are cost shares that are 3-strict for the unmodified primal-dual Steiner forest algorithm.
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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 | 0.004 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 0.001 |
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".