Approximating Unique Games Using Low Diameter Graph Decomposition
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Bibliographic record
Abstract
We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints. Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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