Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling
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Bibliographic record
Abstract
The minimum-weight set cover problem is widely known to be O(log n)-approximable, with no improvement possible in the general case. We take the approach of exploiting problem structure to achieve better results, by providing a geometry-inspired algorithm whose approximation guarantee depends solely on an instance-specific combinatorial property known as shallow cell complexity (SCC). Roughly speaking, a set cover instance has low SCC if any column-induced submatrix of the corresponding element-set incidence matrix has few distinct rows. By adapting and improving Varadarajan’s recent quasi-uniform random sampling method for weighted geometric covering problems, we obtain strong approximation algorithms for a structurally rich class of weighted covering problems with low SCC. We also show how to derandomize our algorithm. Our main result has several immediate consequences. Among them, we settle an open question of Chakrabarty et al. [8] by showing that weighted instances of the capacitated covering problem with underlying network structure have O(1)-approximations. Additionally, our improvements to Varadarajan’s sampling framework yield several new results for weighted geometric set cover, hitting set, and dominating set problems. In particular, for weighted covering problems exhibiting linear (or near-linear) union complexity, we obtain approximability results agreeing with those known for the unweighted case. For example, we obtain a constant approximation for the weighted disk cover problem, improving upon the 2 O(log ∗ n)-approximation known prior to our work and matching the O(1)approximation known for the unweighted variant.
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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 | 0.001 |
| 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