Postman Problems on Mixed Graphs
Bibliographic record
Abstract
The mixed postman problem consists of finding a minimum cost tour of a mixed graph M = (V,E,A) traversing all its edges and arcs at least once. We prove that two well-known linear programming relaxations of this problem are equivalent. The extra cost of a mixed postman tour T is the cost of T minus the cost of the edges and arcs of M. We prove that it is NP-hard to approximate the minimum extra cost of a mixed postman tour. \nA related problem, known as the windy postman problem, consists of finding a minimum cost tour of an undirected graph G=(V,E) traversing all its edges at least once, where the cost of an edge depends on the direction of traversal. We say that G is windy postman perfect if a certain windy postman polyhedron O (G) is integral. We prove that series-parallel undirected graphs are windy postman perfect, therefore solving a conjecture of Win. \nGiven a mixed graph M = (V,E,A) and a subset R ⊆ E ∪ A, we say that a mixed postman tour of M is restricted if it traverses the elements of R exactly once. The restricted mixed postman problem consists of finding a minimum cost restricted tour. We prove that this problem is NP-hard even if R=A and we restrict M to be planar, hence solving a conjecture of Veerasamy. We also prove that it is NP-complete to decide whether there exists a restricted tour even if R=E and we restrict M to be planar. \nThe edges postman problem is the special case of the restricted mixed postman problem when R=A. We give a new class of valid inequalities for this problem. We introduce a relaxation of this problem, called the b-join problem, that can be solved in polynomial time. We give an algorithm which is simultaneously a 4/3-approximation algorithm for the edges postman problem, and a 2-approximation algorithm for the extra cost of a tour. \nThe arcs postman problem is the special case of the restricted mixed postman problem when R=E. We introduce a class of necessary conditions for M to have an arcs postman tour, and we give a polynomial-time algorithm to decide whether one of these conditions holds. We give linear programming formulations of this problem for mixed graphs arising from windy postman perfect graphs, and mixed graphs whose arcs form a forest.
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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 | 0.002 | 0.007 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.028 | 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".