Carving-decomposition based algorithms for the maximum path coloring problem
Bibliographic record
Abstract
Given a set P of paths in a graph G and k colors, the maximum path coloring (Max-PC) problem is to find a maximum subset of P and assign a color to each path of the subset such that the paths with the same color are edge-disjoint. The Max-PC problem is an abstract model for many important routing problems including the all-optical routing. We give a carving-decomposition based exact algorithm for the Max-PC problem. A carving-decomposition of G is a system of edge-cut sets which decomposes G into subgraphs with each vertex of G a minimal subgraph. Our algorithm first finds a carving-decomposition of G and then solves the problem using the dynamic programming based on the carving-decomposition. We also give a 1.58-approximation algorithm for the Max-PC problem. Let L be the maximum number of paths in P on any edge of G and let γ be the maximum cardinality of any edge-cut in a given carving-decomposition. Our exact algorithm solves the Max-PC problem in O((L + 1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.5kγ</sup> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) time and the approximation algorithm runs in O((L + 1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.5γ</sup> kn <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) time for G of n vertices. Our algorithms can be used to solve the Max-PC problem on directed graphs as well. Our computational study shows that the exact algorithm can solve the Max-PC problem for small k and γ in a practical time and the approximation algorithm gives solutions close to the optimal ones for practical values of k and L on graphs with small γ such as rings.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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".