Improved bounds for randomly sampling colorings via linear programming
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Bibliographic record
Abstract
A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k-colorings of a graph G on n vertices with maximum degree Δ is rapidly mixing for k ≥ Δ + 2. In FOCS 1999, Vigoda [43] showed that the flip dynamics (and therefore also Glauber dynamics) is rapidly mixing for any [MATH HERE]. It turns out that there is a natural barrier at [MATH HERE], below which there is no one-step coupling that is contractive with respect to the Hamming metric, even for the flip dynamics. We use linear programming and duality arguments to fully characterize the obstructions to going beyond [MATH HERE]. These extremal configurations turn out to be quite brittle, and in this paper we use this to give two proofs that the Glauber dynamics is rapidly mixing for any [MATH HERE] for some absolute constant ϵe0 > 0. This is the first improvement to Vigoda's result that holds for general graphs. Our first approach analyzes a variable-length coupling in which these configurations break apart with high probability before the coupling terminates, and our other approach analyzes a one-step path coupling with a new metric that counts the extremal configurations. Additionally, our results extend to list coloring, a widely studied generalization of coloring, where the previously best known results required k > 2Δ.
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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 | 0.000 | 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 it