Generalized Cuts and Grothendieck Covers: a Primal-Dual Approximation Framework Extending the Goemans--Williamson Algorithm
Bibliographic record
Abstract
We provide a primal-dual framework for randomized approximation algorithms utilizing semidefinite programming (SDP) relaxations. Our framework pairs a continuum of APX-complete problems including MaxCut, Max2Sat, MaxDicut, and more generally, Max-Boolean Constraint Satisfaction and MaxQ (maximization of a positive semidefinite quadratic form over the hypercube) with new APX-complete problems which are stated as convex optimization problems with exponentially many variables. These new dual counterparts, based on what we call Grothendieck covers, range from fractional cut covering problems (for MaxCut) to tensor sign covering problems (for MaxQ). For each of these problem pairs, our framework transforms the randomized approximation algorithms with the best known approximation factors for the primal problems to randomized approximation algorithms for their dual counterparts with reciprocal approximation factors which are tight with respect to the Unique Games Conjecture. For each APX-complete pair, our algorithms solve a single SDP relaxation and generate feasible solutions for both problems which also provide approximate optimality certificates for each other. Our work utilizes techniques from areas of randomized approximation algorithms, convex optimization, spectral sparsification, as well as Chernoff-type concentration results for random matrices.
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How this classification was reachedexpand
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 | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".