Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, November 15, 2011, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Timon Hertli

Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains

Very recently Thurley [1] gave a randomized algorithm that approximates the number of satisfying assignments of a k-CNF. The running time is still exponential, but considerably faster than the fastest known algorithms counting exactly.

I will present a classical result by Jerrum and Sinclair [2]: If we can approximate the number of satisfying assignments up to a factor of poly(n), we can also do so up to (1+1/poly(n)). This holds more generally for so-called self-reducible problems. The result combines rapidly mixing markov chains with a reduction from approximate counting to almost uniform generation by Jerrum, Valiant and Vazirani [3].

While this result is not necessary for [1], it might be useful for obtaining new approximation algorithms.

[1] Marc Thurley: An Approximation Algorithm for #k-SAT (arXiv, 2011)
[2] Mark R. Jerrum, Alistair Sinclair: Approximate counting, uniform generation and rapidly mixing markov chains (Information and Computation, 1989)
[3] Mark R. Jerrum, Leslie G. Valiant, Vijay V. Vazirani: Random generation of combinatorial structures from a uniform distribution (Theoretical Computer Science, 1986)

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