Sampling and counting Our running example the matching

Sampling and counting Our running example: the matching problem: - Assume know how to sample matchings - How do we count all matchings ? [Section 3. 2]

Partition function [Section 3. 2] Suppose we have a set of states and a function f: R+. Suppose we can sample from a distribution proportional to f, i. e. ¼(!) ® f(!) for every !2. The partition function is: Z = !2 f(!). Example: monomer-dimer model

Sampling and counting Sketch of the counting from sampling approach: [Section 3. 2]

Sampling and counting [Section 3. 1] How to use sampling to (approximately ) count ? Def: A randomized approximation scheme for a counting problem f is a randomized algorithm that, for a given input x and an error tolerance ²>0, outputs a value N s. t. Pr( (1 -²)f(x) · N · (1+²)f(x) ) ¸ ¾ Fully polynomial randomized approximation scheme (FPRAS) is a randomized appx scheme that runs in time polynomial in |x| and 1/². Note: ¾ can be replaced by any constant in (½, 1).

Sampling and counting [Section 3. 2] Proposition 3. 4: Let G be a graph with n vertices and m¸ 1 edges. If there is an approximate sampler for matchings in G with running time T(n, m, ²), then there is an FPRAS counting all matchings in time cm 2²-2 T(n, m, ²/(6 m)) for some constant c>0.

Sampling and counting [Section 3. 2]

Sampling and counting [Section 3. 2]

Sampling and counting [Section 3. 2]

Sampling and counting [Section 3. 2]