A Primal-Dual Extension of the Goemans--Williamson Algorithm for the Weighted Fractional Cut-Covering Problem
Bibliographic record
Abstract
We study a weighted generalization of the fractional cut-covering problem, which we relate to the maximum cut problem via antiblocker and gauge duality. This relationship allows us to introduce a semidefinite programming (SDP) relaxation whose solutions may be rounded into fractional cut covers by sampling via the random hyperplane technique. We then provide a $1/α_{\scriptscriptstyle \mathrm{GW}}$-approximation algorithm for the weighted fractional cut-covering problem, where $α_{\scriptscriptstyle \mathrm{GW}} \approx 0.878$ is the approximation factor of the celebrated Goemans--Williamson algorithm for the maximum cut problem. Nearly optimal solutions of the SDPs in our duality framework allow one to consider instances of the maximum cut and the fractional cut-covering problems as primal-dual pairs, where cuts and fractional cut covers simultaneously certify each other's approximation quality. We exploit this relationship to introduce new combinatorial certificates for both problems, as well as a randomized polynomial-time algorithm for producing such certificates. In~particular, we~show how the Goemans--Williamson algorithm implicitly approximates a weighted instance of the fractional cut-covering problem, and how our new algorithm explicitly approximates a weighted instance of the maximum cut problem. We conclude by discussing the role played by geometric representations of graphs in our results, and by proving our algorithms and analyses to be optimal in several aspects.
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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".