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

2. 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 Bipartite matching and the Hungarian method

4. Bipartite matching and the Hungarian method Nonbipartite matching

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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