Maximum Edge-Disjoint Paths in Planar Graphs with Congestion 2
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Bibliographic record
Abstract
We study the maximum edge-disjoint path problem (MEDP) in planar graphs. We are given a set of terminal pairs and wish to find a maximum routable subset of demands. That is, a subset of demands that can be connected by edge-disjoint paths. It is well-known that there is an integrality gap of order square root of the number of nodes for this problem even on a grid-like graph, and hence in planar graphs (Garg et al.). In contrast, Chekuri et al. show that for planar graphs, if LP is the optimal solution to the natural linear programming relaxation for MEDP, then there is a subset of size OPT over the logarithm of the number of nodes which is routable with congestion 2. Subsequently they showed that it is possible to get within a constant factor of the optimal solution with congestion 4 instead of 2. We strengthen this latter result to show that a constant approximation is possible also with congestion 2 (and this is tight via the integrality gap grid example). We use a basic framework from work by Chekuri et al. At the heart of their approach is a 2-phase algorithm that selects an Okamura-Seymour instance. Each of their phases incurs a factor 2 congestion. It is possible to reduce one of the phases to have congestion 1. In order to achieve an overall congestion 2, however, the two phases must share capacity more carefully. For the Phase 1 problem, we extract a problem called rooted clustering that appears to be an interesting problem class in itself.
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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 | 0.000 | 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