Sampling and counting

1. [Chapter 3] Sampling and counting How do we sample perfect matchings in a planar graph ? [Sampling: uniformly at random, i.e. each matching is equally likely.]

2. [Section 3.1] Sampling and counting The process can be used reversely: using 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).

3. [Section 3.2] Sampling and counting (Modified) Proposition 3.4: Let G be a graph with n vertices and m¸1 edges. If there is an algorithm with running time T(n,m) that produces a uniformly random perfect matching of G, then there is an FPRAS counting all perfect matchings in time cm2²-2T(n,m) for some constant c>0.

4. [Section 3.2] Sampling and counting (Modified) Proposition 3.4: Let G be a graph with n vertices and m¸1 edges. If there is an algorithm with running time T(n,m) that produces a uniformly random perfect matching of G, then there is an FPRAS counting all perfect matchings in time cm2²-2T(n,m) for some constant c>0.

5. [Section 3.2] Sampling and counting (Modified) Proposition 3.4: Let G be a graph with n vertices and m¸1 edges. If there is an algorithm with running time T(n,m) that produces a uniformly random perfect matching of G, then there is an FPRAS counting all perfect matchings in time cm2²-2T(n,m) for some constant c>0.

6. [Section 3.2] Sampling and counting Proof of Proposition 3.4: sketch

7. [Section 3.2] Sampling and counting Proof of Proposition 3.4: sketch

8. [Section 3.2] Sampling and counting Proof of Proposition 3.4: sketch

9. [Section 3.2] Sampling and counting Proof of Proposition 3.4: sketch

10. [Section 3.2] Sampling and counting Proof of Proposition 3.4: sketch

11. [Section 3.1] Sampling and counting Original Prop. 3.4: about approximate (not exact) samplers. Def:Total variation distance of distributions ¼ and ¼’ on the same countable set : ||¼-¼’||TV := ½ !2 |¼(!)-¼’(!)| = maxA½ |¼(A)-¼’(A) | Def:Fully polynomial approximate sampler (FPAUS): given an input x and a sampling tolerance ±>0, the sampler produces an element from a distribution with total variation distance ·± from the target distribution.

12. [Section 3.3] Markov chains • Def: A Markov chain is a sequence (Xt2)t=0,…,1 of random variables with state space , where Xt+1 depends only on Xt. • Example: MC for matchings: •  = ? • X0 = ; • for t¸0: choose a random edge e • if e in Xt, remove it, i.e. Xt+1 = Xt - {e} • if e not in Xt and the endpoints of e are not matched in Xt, then add it, i.e. Xt+1 = Xt [ {e} • otherwise, let Xt+1=Xt

13. [Section 3.3] Markov chains Note: only discrete-time MC on a finite state space  Def: The transition matrix of a MC: an ||x|| matrix P where P(x,y) = Pr(Xt+1=y | Xt=x). Example: MC for matchings:

14. [Section 3.3] Markov chains • Def: A stationary distribution¼:[0,1] satisfies • ¼(y) = x2¼(x)P(x,y) • An MC is • Irreducible if for every x,y there is a t s.t. Pt(x,y)>0 • Aperiodicif gcd{t: Pt(x,x)>0}=1 for every x2 • Ergodic if both irreducible and aperiodic • Thm 3.6: An ergodic MC has a unique stationary distribution ¼; moreover, Pt(x,y) converges to ¼(y) as t1.

15. [Section 3.3] Markov chains Lemma 3.7: If ¼’ satisfies ¼’(x)P(x,y) = ¼’(y)P(y,x) for every x,y2, and x2¼’(x) = 1, Then ¼’ is a stationary distribution of the MC. Note: the condition from Lemma 3.7 is known as detailed balance (or reversibility of the MC).