Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, February 19, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Andrea Francke

On Perfect Matchings, Counting, Sampling, and Self-Reducibility

How do you count structures if you only have time to look at a few of them? If you can sample instances of a 'self-reducible' structure almost uniformly at random and efficiently, you can also efficiently approximate the number of such structures with a fully polynomial approximation scheme (FPRAS) [1,2]. I will illustrate this method by presenting (excerpts of) Jerrum and Sinclair's approximation of the permanent of dense 0-1 matrices [3,4], which at the same time is an approximation of the number of perfect matchings in a bipartite graph. This result was part of Jerrum an Sinclair's seminal work which earned them the 1996 Gödel prize [5]. If time permits, I will also talk about self-reducible structures.

[1] Mark R. Jerrum, Leslie G. Valiant, Vijay V. Vazirani: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43: 169-188 (1986)
[2] Andrei Z. Broder: How hard is to marry at random? (On the approximation of the permanent). STOC 1986: 50-58. See also Erratum STOC 1988: 551
[3] Mark Jerrum, Alistair Sinclair: "Approximate counting, uniform generation and rapidly mixing Markov chains," Information and Computation 82 (1989), 93-133
[4] Mark Jerrum, Alistair Sinclair: "Approximating the permanent," SIAM Journal on Computing 18 (1989), 1149-1178
[5] http://www.sigact.org/Prizes/Godel/1996.html

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