A polynomial-time exact algorithm for the connected k-facility location problem on trees
Bibliographic record
Abstract
This paper studies the Connected [Formula: see text]-Facility Location Problem (Con[Formula: see text]FLP) on trees. Let [Formula: see text] be an undirected tree, where [Formula: see text] is the [Formula: see text]-vertices set and [Formula: see text] is the [Formula: see text]-edges set. A facility set [Formula: see text] and a client set [Formula: see text] are given. Each client [Formula: see text] has one weight [Formula: see text] denoting the demand amount of [Formula: see text], and each facility [Formula: see text] has a weight [Formula: see text] denoting the opening cost at [Formula: see text], and each edge [Formula: see text], for [Formula: see text], is associated with a weight [Formula: see text] denoting the connection cost of it. When some facilities [Formula: see text] are opened, the overall cost involved in Con[Formula: see text]FLP includes three parts: the cost of opening facilities [Formula: see text], [Formula: see text] times the cost of Steiner tree interconnecting all the opened facilities where [Formula: see text] is a fixed parameter, and the total connection cost of assigning each client to the closest facility in [Formula: see text]. The goal of Con[Formula: see text]FLP is to open at most [Formula: see text] facilities to minimize the overall cost, for a given input parameter [Formula: see text]. This paper focuses on the case of Con[Formula: see text]FLP on trees where [Formula: see text], and as a result presents a polynomial-time exact dynamic programming algorithm and a computational experiment to illustrate it. Furthermore, a simple way is shown to adapt the algorithm to the general case of [Formula: see text] and [Formula: see text]. Finally, we apply the algorithm to a Con[Formula: see text]FLP instance in a regional tree-like water transportation network.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.001 | 0.005 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.005 | 0.003 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.022 | 0.006 |
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".