1. Show that the Markov chain used for the MCMC simulation of the hard-core model (Example 1) is

Rachunek Prawdopodobieństwa 2

Zestaw zadań nr 8

Termin realizacji: 16 I 2008

1. Show that the Markov chain used for the MCMC simulation of the hard-core model (Example 1) is

• irreducible

• aperiodic.

2. Show that the Gibbs sampler for random q-colorings (Example 2)

• has ρ

as a stationary distribution,

• is aperiodic, and,

• for q ≥ ∆(G) + 2, it is irreducible.

3. Let G = (V, E) be a connected graph and let a function X be chosen uniformly at random from [q]

. Show that the probability that X is a q-coloring is at most

µ q − 1 q

¶

.

4. Udowodnić, że jeśli 0 ≤ ² ≤ 1, a

, . . . , a

, b

, . . . , b

> 0 i, dla wszystkich j = 1, . . . , k

1 − ² 2k ≤ a

b

≤ 1 + ² 2k , to

1 − ² ≤ Q

j=1

a

Q

j=1

b

≤ 1 + ².

5. (Generalized hard-core model) Given a graph and a parameter λ > 0, define for each configu- ration ξ ∈ {0, 1}

a probability measure

µ

(ξ) = λ

Z

if ξ is feasible and 0 otherwise, where n(ξ) is the number of 1’s in ξ and

Z

= X

(ξ

feasible

λ

is a normalizing constant.

(a) Compute the conditional probability that a vertex v ∈ V takes the value 1, given that all neighbors of v take value 0.

(b) Construct an MCMC algorithm for this generalized hard-core model.

6. The error probability 1/3 in an rptas can be cut down to any given δ > 0 by the following method: Run the algorithm many times (say m, where m is odd), and take the median of the outputs (i.e., the

th largest output). Show that this works for m large enough, and give an explicit bound (depending on δ) for determining how large “large enough” is.

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