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)1.5kγn2) time and the approximation algorithm runs in O((L + 1)1.5γkn2) 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 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 | 0.004 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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".