A New Resource-Constrained Multicommodity Flow Model for Conflict-Free Train Routing and Scheduling
Bibliographic record
Abstract
This paper addresses the problem of generating conflict-free train schedules on a microscopic model of the railway infrastructure. Conflicts arise if two or more trains are scheduled to block the same track section at the same time. A standard model for this problem is the so-called conflict graph, where each considered train path corresponds to a vertex, and edges represent pairwise conflicts so that a conflict-free schedule corresponds to a maximum independent set. Because the linear programming relaxation of the conflict graph formulation is typically very weak, we develop an alternative model using the sequence of resources that each train path passes, encoded in a resource tree. For each resource, we can efficiently determine the maximal conflict cliques by scanning through the blocking times of all train paths and use these cliques as strong cutting planes in an integer linear programming formulation. We show that the number of maximal conflict cliques is linear in the number of train paths, so the ILP formulation uses much fewer but stronger constraints compared to the conflict graph model. In tests with real-world data from the Swiss Federal Railways, the new Resource Tree Conflict Graph model generates for major stations within seconds, even though the underlying model contains about half a million binary variables. This corresponds to a reduction of the computation time of roughly two orders of magnitude when compared to previous approaches and thus allows us to tackle considerable larger problem instances.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".