A (1 + ε)-approximation algorithm for partitioning hypergraphs using a new algorithmic version of the Lovász Local Lemma
Bibliographic record
Abstract
In his seminal result, Beck gave the first algorithmic version of the Lov'asz Local Lemma by giving polynomial time algorithms for 2-coloring and partitioning uniform hypergraphs. His work was later generalized by Alon, and Molloy and Reed. Recently, Czumaj and Scheideler gave an efficient algorithm for 2-coloring non-uniform hypergraphs. But the partitioning algorithm obtained based on their second paper only applies to a more limited range of hypergraphs, so much so that their work doesn't imply the result of Beck for the uniform case. Here we give an algorithmic version of the general form of the Local Lemma which captures (almost) all applications of the results of Beck and Czumaj and Scheideler, with an overall simpler proof. In particular, if H is a non-uniform hypergraph in which every edge e i intersects at most je i j2 other edges of size at most k, for some small constant ff, then we can find a partitioning of H in expected linear time. This result implies the result of Beck for uniform hypergraphs along with a speedup in his running time.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".