A polynomial-time exact algorithm for the connected k-facility location problem on trees
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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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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 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 it