Minimum cycle and homology bases of surface-embedded graphs
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We study the problems of finding a minimum cycle basis (a minimum-weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum-weight set of cycles that generates the 1-dimensional $(\mathbb{Z}_2)$-homology classes) of an undirected graph cellularly embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 1-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^ω + 2^{2g} n^2 + m)$-time algorithm for graphs cellularly embedded on an orientable surface of genus $g$. Prior to this work, the best known algorithms for surface-embedded graphs were those for general graphs: an $O(m^ω)$-time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2m)$-time algorithm. For the minimum homology basis problem, we give a deterministic $O((g + b)^3n\log n + m)$-time algorithm for graphs cellularly embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, improving on existing algorithms for many values of $g$ and $n$. The algorithm assumes that shortest paths are unique; this assumption can be avoided by either using random perturbations of the edge weights guaranteeing a high probability of success or by deterministic means at a cost of an $O(\log n)$ factor increase in running time.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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