Improved Lower Bounds on Multiflow-Multicut Gaps
Bibliographic record
Abstract
Given a set of source-sink pairs, the maximum multiflow problem asks for the maximum total amount of flow that can be feasibly routed between them. The minimum multicut, a dual problem to multiflow, seeks the minimum-cost set of edges whose removal disconnects all the source-sink pairs. It is easy to see that the value of the minimum multicut is at least that of the maximum multiflow, and their ratio is called the multiflow-multicut gap. The classical max-flow min-cut theorem states that when there is only one source-sink pair, the gap is exactly one. However, in general, it is well known that this gap can be arbitrarily large. In this paper, we study this gap for classes of planar graphs and establish improved lower bound results. In particular, we show that this gap is at least 20/9 for the class of planar graphs, improving upon the decades-old lower bound of 2. More importantly, we develop new techniques for proving such a lower bound, which may be useful in other settings as well.
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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 | 0.005 | 0.028 |
| Meta-epidemiology (narrow) | 0.004 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.005 | 0.004 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.006 | 0.014 |
| Open science | 0.007 | 0.008 |
| Research integrity | 0.003 | 0.010 |
| Insufficient payload (model declined to judge) | 0.018 | 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".