Min-Max Latency Walks: Approximation Algorithms for Monitoring Vertex-Weighted Graphs
Bibliographic record
Abstract
In this paper, we consider the problem of planning a path for a robot to monitor a known set of features of interest in an environment.We represent the environment as a vertex- and edge-weighted graph, where vertices represent features or regions of interest. The edge weights give travel times between regions, and the vertex weights give the importance of each region. If the robot repeatedly performs a closed walk on the graph, then we can define the latency of a vertex to be the maximum time between visits to that vertex, weighted by the importance (vertex weight) of that vertex. Our goal in this paper is to find the closed walk that minimizes the maximum weighted latency of any vertex. We show that there does not always exist an optimal walk of polynomial size. We then prove that for any graph there exist a constant approximation walk of size O(n 2), where n is the number of vertices. We provide two approximation algorithms; an O(log n)-approximation and an O(log ρ)-approximation, where ρ is the ratio between the maximum and minimum vertex weight. We provide simulation results which demonstrate that our algorithms can be applied to problems consisting of thousands of vertices.
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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 | 0.000 | 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 | 0.000 | 0.002 |
| 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 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".