Short cycles via low-diameter decompositions
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Bibliographic record
Abstract
We present improved algorithms for short cycle decomposition of a graph - a decomposition of an undirected, unweighted graph into edge-disjoint cycles, plus a small number of additional edges. Short cycle decompositions were introduced in the recent work of Chu et al. (FOCS 2018), and were used to make progress on several questions in graph sparsification.For all constants δ ∈ (0,1], we give an O(mnδ) time algorithm that, given a graph G, partitions its edges into cycles of length [MATH HERE], with O(n) extra edges not in any cycle. This gives the first subquadratic, in fact almost linear time, algorithm achieving polylogarithmic cycle lengths. We also give an m · [MATH HERE] time algorithm that partitions the edges of a graph into cycles of length [MATH HERE], with O(n) extra edges not in any cycle. This improves on the short cycle decomposition algorithms given by Chu et al. in terms of all parameters, and is significantly simpler.As a result, we obtain faster algorithms and improved guarantees for several problems in graph sparsification - construction of resistance sparsifiers, graphical spectral sketches, degree preserving sparsifiers, and approximating the effective resistances of all edges.
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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 | 0.000 | 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 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it