FACES OF MATCHING POLYHEDRA
Bibliographic record
Abstract
Let G = (V, E, ~) be a finite loopless graph, let \nb=(bi:ieV) be a vector of positive integers. A \nfeasible matching is a vector X = (x.: j e: E) \nJ \nof nonnegative \nintegers such that for each node i of G, the sum of the \nover the edges j of G incident with i is no \ngreater than bi. The matching polyhedron P(G, b) is the \nconvex hull of the set of feasible matchings. \nIn Chapter 3 we describe a version of Edmonds' blossom \nalgorithm which solves the problem of maximizing C • X \nover P (G, b) where c =. (c.: j e: E) \nJ \nis an arbitrary real \nvector. This algorithm proves a theorem of Edmonds which \ngives a set of linear inequalities sufficient to define \nP(G, b). \nIn Chapter 4 we prescribe the unique subset of these \ninequalities which are necessary to define P(G, b), that \nis, we characterize the facets of P(G, b). We also \ncharacterize the vertices of P(G, b), thus describing the \nstructure possessed by the members of the minimal set X \nof feasible matchings of G such that for any real vector \nc = (c.: j e: E), c • x is maximized over P(G, b) \nJ \nmember of X. \nby a \nIn Chapter 5 we present a generalization of the blossom \nalgorithm which solves the problem: maximize c • x over \na face F of P(G, b) for any real vector c = (c.: j e: E). \nJ \nIn other words, we find a feasible matching x of G which \nsatisfies the constraints obtained by replacing an arbitrary \nsubset of the inequalities which define P(G, b) by equations and which maximizes c • x subject to this \nrestriction. We also describe an application of this \nalgorithm to matching problems having a hierarchy of objective \nfunctions, so called ''multi-optimization'' problems. \nIn Chapter 6 we show how the blossom algorithm can be \ncombined with relatively simple initialization algorithms \nto give an algorithm which solves the following postoptimality \nproblem. Given that we know a matching 0 x £ P(G, b) \nmaximizes c · x over P(G, b), we wish to utilize 0 \nX \nwhich \nto \nfind a feasible matching x' £ P(G, b') which maximizes \nc • x over P(G, b'), where b' = (b!: i £ V) \n]_ \nvector of positive integers and \narbitrary real vector. \nc=(c.:j£E) \nJ \nis a \nis an \nIn Chapter 7 we describe a computer implementation of \nthe blossom algorithm described herein.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.025 | 0.003 |
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".