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Exploring Matching Theory: From Bipartite Matching to Randomized Algorithms

This article delves into matching theory, covering topics such as bipartite matching, Hungarian method, alternating paths, linear programming, determinants, and randomized algorithms, with a focus on polyhedral combinatorics. It discusses key theorems, algorithms, and structures in matching theory, including the min-max theorem, Ford-Fulkerson algorithm, and total unimodularity. Additionally, it explores the applications of matching theory in various areas like matroids, stable sets, and planar graphs, highlighting the significance of randomized algorithms in solving complex matching problems. The text also touches on sampling techniques, Markov chain sampling, and rapid mixing proofs, emphasizing their role in optimizing solutions and generating random witnesses. With a blend of theoretical concepts and practical applications, this article offers insights into the diverse realms of matching theory.

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Exploring Matching Theory: From Bipartite Matching to Randomized Algorithms

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  1. Matchings and where they lead us László Lovász Microsoft Research lovasz@microsoft.com

  2. Bipartite matching and the Hungarian method Existence and min-max Theorem: Frobenius 1912, 1917 König 1915, 1931 Egerváry 1931 Polynomial time algorithm: Kuhn 1955 Structure theory: König 1916 Dulmage-Mendelssohn 1958-59

  3. Nonbipartite matching Bipartite matching and the Hungarian method

  4. Nonbipartite matching Existence and min-max Theorem: Tutte 1947 Polynomial time algorithm: Edmonds 1965 Structure theory: Gallai 1963 Edmonds 1965 Kotzig 1959-60

  5. Alternating paths Bipartite matching and the Hungarian method Nonbipartite matching

  6. Alternating paths Maximum flow: Ford-Fulkerson 1956 Matroid intersection: Edmonds 1969 Matroid matching: Lovász 1980 Stable sets in claw-free graphs: Minty, Sbihi 1980 Jump systems: Boucher, Cunningham 1995

  7. Linear programming Bipartite matching and the Hungarian method Nonbipartite matching Alternating paths

  8. Linear programming Total unimodularity: Hoffman-Kuhn 1956 Hoffman-Kruskal 1956 Total dual integrality: Hoffman 1970 Edmonds-Giles 1977 Perfect graphs: Berge 1959 Fulkerson 1971 Lovász 1972

  9. Linear programming Polyhedral combinatorics Bipartite matching and the Hungarian method Nonbipartite matching Alternating paths

  10. Polyhedral combinatorics The matching polytope: Edmonds 1965 Equivalence of separation and optimization: Grötschel-Lovász-Schrijver 1981

  11. Linear programming Polyhedral combinatorics Determinants Bipartite matching and the Hungarian method Nonbipartite matching Alternating paths

  12. Determinants Determinants vs. bipartite perfect matchings: König 1915 Determinants vs. non-bipartite perfect matchings: Tutte 1947 Linear algebra and bipartite matching: Perfect 1966 Edmonds 1967 Generic rigidity, geometric representations,...

  13. Linear programming Polyhedral combinatorics Randomized algorithms Bipartite matching and the Hungarian method Nonbipartite matching Alternating paths Determinants

  14. Randomized algorithms (just substitute) Bipartite graphs: Edmonds 1967 Running time analysis: Schwarz 1978 Lovász 1979 Exact matching:only by random substitution!

  15. Linear programming Polyhedral combinatorics Counting Bipartite matching and the Hungarian method Nonbipartite matching Alternating paths Determinants Randomized algorithms

  16. Substitute +1 or -1 to compensate for the sign? (Polya) Substitute randomly and take expectation?

  17. Counting Planar graph: Kasteleyn 1957 Characterization: McCuaig-Robertson -Seymour-Thomas 1997 Approximate, determinants: Godsil-Gutman 1980 Alternating path strikes back When is the variance small? Approximate, by sampling: Jerrum-Sinclair 1988

  18. Counting: number of witnesses Existence: language in NP Sampling: generate random witness Optimization: find optimal witness Four problems for NP Property in NP: Witness of a property in NP:

  19. Linear programming Polyhedral combinatorics Sampling Bipartite matching and the Hungarian method Nonbipartite matching Alternating paths Determinants Counting Randomized algorithms

  20. model for sampling from knapsack solutions, contingency tables, convex bodies, eulerian orientations,... Sampling Markov chain sampling: Broder 1986 Rapid mixing proof (dense graphs): Jerrum-Sinclair 1988 Extension to non-dense, bipartite: Jerrum-Sinclair-Vigoda 2002

  21. often gets perfect matching takes long to get perfect matching + mixes in poly time How about non-bipartite non-dense graphs?

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