Parameterized Approximation Algorithms for K-center Clustering and Variants
Bibliographic record
Abstract
k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm yields a 2^{O((klog k)/{epsilon})}dn-time (1+epsilon)-approximation for Euclidean k-center, where d is the dimension. In this work, we give a faster algorithm for small dimensions: roughly speaking an O^*(2^{O((1/epsilon)^{O(d)} k^{1-1/d} log k)})-time (1+epsilon)-approximation. In particular, the running time is roughly O^*(2^{O((1/epsilon)^{O(1)}sqrt{k}log k)}) in the plane. We complement our algorithmic result with a matching hardness lower bound. We also consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2^{O(klog k)}n^2 time 3-approximation for NUkC, and a 2^{O((klog k)/epsilon)}dn time (1+\epsilon)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.
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How this classification was reachedexpand
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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".