A scalable exact algorithm for the vertex \(p\)-center problem
Bibliographic record
Abstract
Abstract The vertex p-center problem consists in selecting p centers among a finite set of candidates and assigning a set of clients to them, with the aim of minimizing the maximum dissimilarity between a client and its associated center. State-of-the-art algorithms for the problem are based on the solution of a series of covering subproblems, and are particularly efficient when additional reduction rules are used to limit the size of the subproblems. In fact, these algorithms do not scale well when the cardinality of the dissimilarity matrix grows above a few million entries, as the time and space required to compute and store the matrix start dominating those required for solving the subproblems. We introduce a scalable relaxation-based iterative algorithm that does not rely on the computation of the entire matrix, but rather relies on the computation of a typically much smaller sub-matrix that is only enlarged if deemed necessary. This method can solve to proven optimality p-center problems derived from the TSP library and containing up to one million clients for small but realistic values of p.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.014 | 0.003 |
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 source (direct Gemma or distilled Codex), 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".