Approximation schemes for Min-Sum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si35.svg" display="inline" id="d1e488"><mml:mi>k</mml:mi></mml:math>-Clustering
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Bibliographic record
Abstract
We consider the Min-Sum k -Clustering ( k -MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G = ( V , E ) and a parameter k , the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized. The k -MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of O ( log | V | ) in polynomial time for general metrics and an approximation ratio 2 + ϵ in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k -MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k -MSC to the balanced k -median problem and design approximation algorithms for the latter to obtain an approximation for k -MSC. In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k -MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k -median and is a trade off between center-based clustering problems (such as balanced k -median) and pair-wise clustering (such as Min-Sum k -clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.
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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 | 0.001 | 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