Multicommodity demand flow in a tree and packing integer programs
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Bibliographic record
Abstract
We consider requests for capacity in a given tree network T = ( V , E ) where each edge e of the tree has some integer capacity u e . Each request f is a node pair with an integer demand d f and a profit w f which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provides a maximum profit. This generalizes well-known problems, including the knapsack and b -matching problems. When all demands are 1, we have the integer multicommodity flow problem. Garg et al. [1997] had shown that this problem is NP-hard and gave a 2-approximation algorithm for the cardinality case (all profits are 1) via a primal-dual algorithm. Our main result establishes that the integrality gap of the natural linear programming relaxation is at most 4 for the case of arbitrary profits. Our proof is based on coloring paths on trees and this has other applications for wavelength assignment in optical network routing. We then consider the problem with arbitrary demands. When the maximum demand d max is at most the minimum edge capacity u min , we show that the integrality gap of the LP is at most 48. This result is obtained by showing that the integrality gap for the demand version of such a problem is at most 11.542 times that for the unit-demand case. We use techniques of Kolliopoulos and Stein [2004, 2001] to obtain this. We also obtain, via this method, improved algorithms for line and ring networks. Applications and connections to other combinatorial problems are discussed.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it