Approximating Unique Games Using Low Diameter Graph Decomposition
Bibliographic record
Abstract
We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition. For the ${\sf Max2Lin}_k$ problem in $K_r$-minor free graphs, when there is an assignment satisfying $1-\varepsilon$ fraction of constraints, we present an algorithm that produces an assignment satisfying $1-O(r\varepsilon)$ fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the ${\sf MaxCut}$ 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 $1-O(r \sqrt{\varepsilon})$ 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 ${\sf Max2Lin}_k$ problem and by $\sqrt{\log g}$ in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the ${\sf Max2Lin}_k$ case, and the latter result generalizes the result of Trevisan for general Unique Games.
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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 | 0.001 | 0.009 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.003 |
| 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".