On Linear Programming Relaxations for Unsplittable Flow in Trees
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Bibliographic record
Abstract
We study some linear programming relaxations for the Unsplittable Flow problem on trees (UFP-Tree). Inspired by results obtained by Chekuri, Ene, and Korula for Unsplittable Flow on paths (UFP-Path), we present a relaxation with polynomially many constraints that has an integrality gap bound of O(log n * min(log m, log n)) where n denotes the number of tasks and m denotes the number of edges in the tree. This matches the approximation guarantee of their combinatorial algorithm and is the first demonstration of an efficiently-solvable relaxation for UFP-Tree with a sub-linear integrality gap. The new constraints in our LP relaxation are just a few of the (exponentially many) rank constraints that can be added to strengthen the natural relaxation. A side effect of how we prove our upper bound is an efficient O(1)-approximation for solving the rank LP. We also show that our techniques can be used to prove integrality gap bounds for similar LP relaxations for packing demand-weighted subtrees of an edge-capacitated tree. On the other hand, we show that the inclusion of all rank constraints does not reduce the integrality gap for UFP-Tree to a constant. Specifically, we show the integrality gap is Omega(sqrt(log n)) even in cases where all tasks share a common endpoint. In contrast, intersecting instances of UFP-Path are known to have an integrality gap of O(1) even if just a few of the rank 1 constraints are included. We also observe that applying two rounds of the Lovász-Schrijver SDP procedure to the natural LP for UFP-Tree derives an SDP whose integrality gap is also O(log n * min(log m, log n)).
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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 | 0.002 |
| 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