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| matching (graph theory) | |
|---|---|
| Name | Matching (graph theory) |
| Field | Graph theory |
matching (graph theory) A matching in a graph is a set of edges without common endpoints; it is a central object studied in combinatorics, discrete mathematics, and theoretical computer science. The concept connects classical results and problems investigated by figures and institutions such as Kőnig's theorem contributors, the Erdős–Rényi model researchers, the Royal Society affiliates, the Hopcroft–Karp algorithm developers, and the Princeton University school of graph theorists. It underpins foundational work by Paul Erdős, Alfréd Rényi, Dénes Kőnig, László Lovász, and Miklós Simonovits and appears in contexts ranging from the Stanford University seminars to the International Mathematical Olympiad.
A matching is formally defined on a finite undirected graph G = (V, E) as a subset M ⊆ E such that no two edges in M share a common vertex; this definition was formalized in texts by Claude Berge and surveys published by the American Mathematical Society. Fundamental invariants include the matching number ν(G) and the size of a maximum matching, studied in classical results by Gallai and contributors associated with the Mathematical Association of America. Properties such as augmenting paths and alternating cycles were highlighted in work by Jack Edmonds and appear alongside parity arguments used by researchers at University of Cambridge and Harvard University. Important theorems include results analogous to those by Kőnig and extensions by Tutte, which relate matchings to factors and coverings discussed in the literature of the London Mathematical Society.
Several specialized matchings have been named and analyzed by mathematical communities and authors including László Lovász and collaborators at the Microsoft Research labs. Examples include: - Perfect matchings, where every vertex is incident to exactly one edge of the matching; classical existence criteria appear in work linked to Tutte's theorem and explorations by W. T. Tutte. - Maximum matchings, the largest possible matchings studied in algorithmic papers from Bell Labs and AT&T. - Maximal matchings, which cannot be extended by adding more edges, discussed in combinatorial expositions from MIT Press and seminars at ETH Zurich. - Induced matchings, strong matchings, and fractional matchings treated in articles appearing in journals affiliated with the Institute of Mathematical Statistics and researchers like Miklós Ajtai and Noga Alon. These types are connected to classic problems and competitions hosted by International Congress of Mathematicians and elaborated in monographs by Cambridge University Press.
Algorithmic treatments trace back to matchings algorithms developed by practitioners at Bell Labs, the Princeton University group, and algorithm designers such as Jack Edmonds and Richard Karp. Polynomial-time algorithms include augmenting path methods, blossom algorithms by Jack Edmonds and improvements like the Micali-Vazirani algorithm developed in research communities at institutions such as University of California, Berkeley. The Hopcroft–Karp algorithm for bipartite graphs, formulated at Columbia University and referenced in the curricula of Massachusetts Institute of Technology, runs in O(√V E). Complexity-theoretic hardness results connect to reductions studied by Stephen Cook and Leonid Levin and to NP-completeness proofs disseminated at conferences of the Association for Computing Machinery. Parameterized and approximation algorithms are topics in proceedings of the European Symposium on Algorithms and work by researchers affiliated with Max Planck Institute and INRIA.
Linear programming formulations for matchings were pioneered in studies by Gérard Cornuéjols and collaborators, with polyhedral descriptions influenced by work at the Institute for Operations Research and the Management Sciences and the Royal Society. The matching polytope, facets, and integrality properties were characterized in seminal papers by Jack Edmonds and expanded by scholars at Columbia University and Rutgers University. Fractional matchings and LP duality relate to coverings and packing theorems explored in texts published by Springer and seminars at University of Oxford. Connections to matroid theory and polyhedral combinatorics were developed by researchers like Jaroslav Nešetřil and groups at the University of Toronto.
Matchings appear across applied and theoretical domains studied by institutions such as Bell Labs, IBM Research, and Google Research. Applications include assignment problems in operations research (classical formulations studied at the London School of Economics), resource allocation in economics (work by Paul Samuelson and Kenneth Arrow influenced related markets), network design in telecommunication projects undertaken by Nokia and AT&T, computational biology collaborations at Broad Institute and Wellcome Trust, and scheduling problems addressed in publications from SIAM conferences. Matchings also underpin auction design analyzed at workshops at Stanford Graduate School of Business and matching markets literature influenced by Alvin E. Roth and Lloyd Shapley including their Nobel-recognized work.
Generalizations include b-matchings, factors, hypergraph matchings, and matroid intersection, topics developed in papers from Princeton University, ETH Zurich, and the University of Waterloo. Hypergraph matching problems intersect with combinatorial set theory work by Paul Erdős and computational complexity investigations led by groups at Carnegie Mellon University. Related structures include vertex covers, edge covers, and independent sets treated in monographs by Cambridge University Press and research networks such as the Fields Institute. Advanced connections involve rook theory studied in combinatorics seminars at Queen Mary University of London and statistical physics parallels explored by researchers at Institute for Advanced Study.