Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications
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Bibliographic record
Abstract
We consider the matroid median problem [Krishnaswamy et al. 2011], wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation in Krishnaswamy et al. [2011]. We illustrate the power and versatility of our techniques by deriving (a) an 8-approximation for the two-matroid median problem, a generalization of matroid median that we introduce involving two matroids; and (b) a 24-approximation algorithm for matroid median with penalties , which is a vast improvement over the 360-approximation obtained in Krishnaswamy et al. [2011]. We show that a variety of seemingly disparate facility-location problems considered in the literature—data placement problem, mobile facility location, k -median forest, metric uniform minimum-latency Uncapacitated Facility Location (UFL)—in fact reduce to the matroid median or two-matroid median problems, and thus obtain improved approximation guarantees for all these problems. Our techniques also yield an improvement for the knapsack median problem.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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