高级算法 (Fall 2020)/Problem Set 4

Problem 1

Given an undirected graph [math]\displaystyle{ G(V, E) }[/math] with maximum degree [math]\displaystyle{ \Delta }[/math], where the vertex set [math]\displaystyle{ V }[/math] is partitioned into [math]\displaystyle{ r }[/math] disjoint subsets, i.e., [math]\displaystyle{ V = S_1 \cup S_2 \cup \cdots \cup S_r }[/math] with [math]\displaystyle{ S_i \cap S_j = \varnothing }[/math] for all [math]\displaystyle{ i \neq j }[/math]. We call a vertex set [math]\displaystyle{ T }[/math] is a transversal of the partition [math]\displaystyle{ \{S_1, S_2, \cdots, S_r\} }[/math], if [math]\displaystyle{ |T \cap S_i| = 1 }[/math] for all [math]\displaystyle{ 1 \leq i \leq r }[/math]. Assume that [math]\displaystyle{ |S_i| \geq 2e\Delta }[/math] for all [math]\displaystyle{ 1 \leq i \leq r }[/math].

1. Prove that there must be an independent transversal (a transversal which is also an independent set) of [math]\displaystyle{ \{S_1, S_2, \cdots, S_r\} }[/math].

Hint: Lovász Local Lemma.

2. Design a randomized algorithm that finds an independent transversal of [math]\displaystyle{ \{S_1, S_2, \cdots, S_r\} }[/math] in expected polynomial time.

Problem 2

A set of vertices [math]\displaystyle{ D\subseteq V }[/math] of graph [math]\displaystyle{ G(V,E) }[/math] is a dominating set if for every [math]\displaystyle{ v\in V }[/math], it holds that [math]\displaystyle{ v\in D }[/math] or [math]\displaystyle{ v }[/math] is adjacent to a vertex in [math]\displaystyle{ D }[/math]. The problem of computing minimum dominating set is NP-hard.

1. Prove that for every [math]\displaystyle{ d }[/math]-regular graph with [math]\displaystyle{ n }[/math] vertices, there exists a dominating set with size at most [math]\displaystyle{ \frac{n(1+\ln(d+1))}{d+1} }[/math].

Hint: Probabilistic method.

2. Try to obtain an upper bound for the size of dominating set using Lovász Local Lemma. Is it better or worse than the previous one?

Problem 3

Let [math]\displaystyle{ G=(V,E) }[/math] be a simple and undirected graph. The Ising model is a distribution [math]\displaystyle{ \mu }[/math] over [math]\displaystyle{ \{-1,+1\}^V }[/math] such that

[math]\displaystyle{ \forall \sigma \in \{-1,+1\}^V, \quad \mu(\sigma) = \frac{1}{Z}\exp\left( -\sum_{\{u,v\}\in E}\beta\sigma(u)\sigma(v) \right), }[/math]

where the parameter [math]\displaystyle{ \beta \in \mathbb{R} }[/math] is called the inverse temperature and the partition function [math]\displaystyle{ Z }[/math] is defined as

[math]\displaystyle{ Z = \sum_{\tau \in \{-1,+1\}^V} \exp\left( -\sum_{\{u,v\}\in E}\beta\tau(u)\tau(v) \right). }[/math]

Hence, [math]\displaystyle{ \mu }[/math] is a joint distribution of [math]\displaystyle{ |V| }[/math] random variables and each variable [math]\displaystyle{ v \in V }[/math] takes its value from [math]\displaystyle{ \{-1,+1\} }[/math].

Let [math]\displaystyle{ \sigma \in \{-1,+1\}^V }[/math] and [math]\displaystyle{ v \in V }[/math]. Let [math]\displaystyle{ \mu_v(\cdot\mid \sigma( V \setminus \{v\})) }[/math] denote the marginal distribution on [math]\displaystyle{ v }[/math] conditioning on the value of [math]\displaystyle{ u }[/math] is fixed as [math]\displaystyle{ \sigma(u) }[/math] for all [math]\displaystyle{ u\neq v }[/math].

The Glauber dynamics for Ising model is defined as follows:

• initially, start from an arbitrary [math]\displaystyle{ X \in \{-1,+1\}^{V} }[/math];
• in each step, pick a vertex [math]\displaystyle{ v \in V }[/math] uniformly at random, and resample [math]\displaystyle{ X(v) }[/math] according to the distribution [math]\displaystyle{ \mu_v(\cdot\mid X( V \setminus \{v\})) }[/math].

Here are the problems.

1. Calculate [math]\displaystyle{ \mu_v(+1\mid \sigma( V \setminus \{v\})) }[/math] and [math]\displaystyle{ \mu_v(-1\mid \sigma( V \setminus \{v\})) }[/math].

Here [math]\displaystyle{ \mu_v(+1\mid \sigma( V \setminus \{v\})) }[/math] is the probability that [math]\displaystyle{ v }[/math] takes the value [math]\displaystyle{ +1 }[/math] conditioning on the value of [math]\displaystyle{ u }[/math] is fixed as [math]\displaystyle{ \sigma(u) }[/math] for all [math]\displaystyle{ u\neq v }[/math].

2. Show that the Glauber dynamics for Ising model is irreducible, aperiodic and reversible with respect to [math]\displaystyle{ \mu }[/math].