随机算法 (Spring 2013)/Problem Set 4

Problem 1

Consider the Markov chain of graph coloring

Markov Chain for Graph Coloring

Start with a proper coloring of [math]\displaystyle{ G(V,E) }[/math]. At each step: Pick a vertex [math]\displaystyle{ v\in V }[/math] and a color [math]\displaystyle{ c\in[q] }[/math] uniformly at random. Change the color of [math]\displaystyle{ v }[/math] to [math]\displaystyle{ c }[/math] if the resulting coloring is proper; do nothing if otherwise.

Show that the Markov chain is:

1. aperiodic;
2. irreducible if [math]\displaystyle{ q\ge \Delta+2 }[/math];
3. with uniform stationary distribution.

Problem 2

Consider the following random walk on hypercube:

Yet another random Walk on Hypercube

At each step, for the current state [math]\displaystyle{ x\in\{0,1\}^n }[/math]: pick an [math]\displaystyle{ i\in\{0,1,2,\ldots,n\} }[/math] uniformly at random; flip [math]\displaystyle{ x_i }[/math] (let [math]\displaystyle{ x_i=1-x_i }[/math]) if [math]\displaystyle{ i\neq 0 }[/math].

• Show that the random walk is ergodic.
• Give the stationary distribution of the random walk.
• Analyze the mixing time of the random walk by coupling.

Hint.1: Construct a coupling rule such that the Hamming distance between two states never increases, and decreases much faster than the naive coupling which forces the two chains to use the same [math]\displaystyle{ i }[/math].

Hint.2: When constructing the coupling, consider using a cyclic permutation of disagreeing positions.

Problem 3

Consider the following random walk over all subsets [math]\displaystyle{ S\in{[n]\choose k} }[/math] for some [math]\displaystyle{ k\le \frac{n}{2} }[/math]:

Random walk over [math]\displaystyle{ k }[/math]-subsets

At each step, for the current state [math]\displaystyle{ S\in{[n]\choose k} }[/math]: with probability [math]\displaystyle{ p }[/math], do nothing; else pick an [math]\displaystyle{ x\in S }[/math] and a [math]\displaystyle{ y\in[n]\setminus S }[/math] independently and uniformly at random, and change the current set to be [math]\displaystyle{ S\setminus\{x\}\cup\{y\} }[/math].

You are allowed to choose a self-loop probability [math]\displaystyle{ p }[/math] for your convenience.

• Show that the random walk is ergodic
• Give the stationary distribution of the random walk.
• Prove that the mixing time is [math]\displaystyle{ O(k\log k) }[/math] by coupling.

Hint.1: Considering a coupling [math]\displaystyle{ (S,T) }[/math], the [math]\displaystyle{ [n] }[/math] is partitioned into [math]\displaystyle{ S\cap T,S\setminus T,T\setminus S,\overline{S\cup T} }[/math]. Design a coupling rule to adopt different cases (regarding which parts the random [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] belong to) so that the difference between two states never increases.

Hint.2: Use a cyclic permutation (with some desirable property) of elements in the symmetric difference [math]\displaystyle{ S\oplus T }[/math] to achieve the above goal.

Problem 4

Let [math]\displaystyle{ G(V,E) }[/math] be a connected undirected simple graph (no self-loops and parallel edges) defined on [math]\displaystyle{ n }[/math] vertices. Let [math]\displaystyle{ \phi(G) }[/math] be the expansion ratio of [math]\displaystyle{ G }[/math]. [math]\displaystyle{ G }[/math] is NOT necessarily regular. For any [math]\displaystyle{ v\in V }[/math], let [math]\displaystyle{ d_v }[/math] be the degree of vertex [math]\displaystyle{ v }[/math].

• What is the largest possible value for [math]\displaystyle{ \phi(G) }[/math]? Construct a graph [math]\displaystyle{ G }[/math] with this expansion ratio and explain why it is the largest.
• What is the smallest possible value for [math]\displaystyle{ \phi(G) }[/math]? Construct a graph [math]\displaystyle{ G }[/math] with this expansion ratio and explain why it is the smallest.
• Run a lazy random walk on [math]\displaystyle{ G }[/math]. What is the stationary distribution? Starting from an arbitrary vertex in an arbitrary unknown [math]\displaystyle{ G }[/math], how long in the worst case should you run the random walk to guarantee the distribution of the current position is within a total variation distance of [math]\displaystyle{ \epsilon }[/math] from the stationary distribution? Give an upper bound of the time in terms of [math]\displaystyle{ n }[/math] and [math]\displaystyle{ \epsilon }[/math].
• Suppose that the maximum degree of [math]\displaystyle{ G }[/math] is known but the graph is not necessarily regular. Design a random walk on [math]\displaystyle{ G }[/math] with uniform stationary distribution. How long should you run the random walk to be within [math]\displaystyle{ \epsilon }[/math]-close to the uniform distribution in the worst case?