Approximation Algorithms for Clustering Problems
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Bibliographic record
Abstract
In this thesis, we present some approximation algorithms for the following clustering problems: Minimum Sum of Radii (MSR), Minimum Sum of Diameters (MSD), and Unsplittable Capacitated Facility Location. Given a metric (V, d) and an integer k, we consider the problem of partitioning the points of V into k clusters so as to minimize the sum of radii (MSR) or the sum of diameters (MSD) of these clusters. We call a cluster containing a single point, a singleton cluster. For the MSR problem when singleton clusters are not allowed, we give an exact algorithm for metrics induced by unweighted graphs. For the MSD problem on the plane with Euclidean distances, we present a polynomial time approximation scheme. In addition, we settle the complexity of the MSD problem with constant $k$ by giving a polynomial time exact algorithm in this case. In the (uniform) UCFL problem, we are given a set of clients and a set of facilities where client j has demand d_j, each facility i has capacity u and opening cost f_i, and a metric cost c_{ij} which denotes the cost of serving one unit of demand of client j at facility i. The goal is to open a subset of facilities and assign each client to exactly one open facility so that the total amount of demand assigned to each open facility is no more than $u$, while minimizing the total cost of opening facilities and serving clients. As it is NP-hard to give a solution without violating the capacities, we consider bicriteria (\alpha,\eta)-approximation algorithms, where these algorithms return a solution whose cost is within factor \alpha of the optimum and violates the capacity constraints within factor \eta. We present the first constant approximations with violation factor less than 2. In addition, we present a quasi-polynomial time (1+\epsilon,1+\epsilon)-approximation for the (uniform) UCFLP in Euclidean metrics, for any constant \epsilon > 0.
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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 | 0.003 |
| 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