Balanced k-Colorings
Bibliographic record
Abstract
While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference between the size of any two color classes in that subset. The discrepancy is the minimum possible imbalance. We show that the discrepancy is always at most 4d \\Gamma 3, where d (the "dimension") is the maximum number of subsets containing a common vertex. For 2-colorings, the bound on the discrepancy is at most maxf2d \\Gamma 3; 2g. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete. Key words: Discrepancy, Balance Theorem, NP completeness Corresponding author Email addresses: biedl@uwaterloo.ca (Therese C. Biedl), ewcenek@uwaterloo.ca (Eowyn Cenek), tmchan@uwaterloo.ca (Timothy M. Chan), edemaine@mit.edu (Erik D. Demaine), mldemaine@uwaterloo.ca (Martin L. Demaine), rudolf@cs.ust.hk (Rudolf Fleischer), m2wang@uwaterloo.ca (Ming-Wei Wang). 1 The research was mainly done while the author was a PhD student at the University of Waterloo, Department of Computer Science 2 The research was mainly done while the author was a visiting associate professor at the University of Waterloo, Department of Computer Science Preprint submitted to Elsevier Science 28 September 2001 1
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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".