Reformulations by Discretization for Piecewise Linear Integer Multicommodity Network Flow Problems
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Bibliographic record
Abstract
We consider the piecewise linear multicommodity network flow problem with the addition of a constraint specifying that the total flow on each arc must be an integer. This problem has applications in transportation and logistics, where total flows might represent vehicles or containers filled with different products. We introduce formulations that exploit this integrality constraint by adapting to our problem a technique known as discretization that has been used to derive mixed-integer programming models for several combinatorial optimization problems. We enhance the discretized models either by adding valid inequalities derived from cut-set inequalities or by using flow disaggregation techniques. Since the size of the formulations derived from discretization and flow disaggregation rapidly increases with problem dimensions, we develop an efficient and effective Lagrangian relaxation method to compute lower and upper bounds. We perform computational results on a large set of randomly generated instances that allow us to compare the relative efficiency of the different modeling alternatives (flow disaggregation plus addition of cut-set inequalities with or without discretization), when used within the Lagrangian relaxation approach.
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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 | 0.000 | 0.001 |
| 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