View source for 高级算法 (Fall 2022)/Problem Set 2

== Problem 1 ==
Suppose we want to estimate the value of  <math>Z</math>. Let  <math>\mathcal{A}</math> be an algorithm that outputs <math>\widehat{Z}</math> satisfying
<math>\Pr[ (1-\epsilon)Z \leq \widehat{Z} \leq (1+\epsilon )Z] \geq \frac{3}{4} .</math>

We run  <math>\mathcal{A}</math> independently for <math>s</math> times, and obtain the outputs <math>\widehat{Z}_1,\widehat{Z}_2,\ldots,\widehat{Z}_s</math>.

Let <math>X</math> be the '''median''' (中位数) of <math>\widehat{Z}_1,\widehat{Z}_2,\ldots,\widehat{Z}_s</math>. Find the number <math>s</math> such that <math>\Pr[ (1-\epsilon)Z \leq X \leq (1+\epsilon )Z] \geq 1 - \delta .</math>

Express <math>s</math> as a function of <math>\delta</math>. Make <math>s</math> as small as possible.

'''Remark''': in this problem, we boost the probability of success from <math>\frac{3}{4}</math> to <math>1-\delta</math>. This method is called the median trick.

'''Hint''': Chernoff bound.

== Problem 2 ==
Let <math>X</math> be a random variable with expectation <math>0</math> such that moment generating function <math>\mathbf{E}[\exp(t|X|)]</math> is finite for some <math> t > 0 </math>. We can use the following two kinds of tail inequalities for <math> X </math>.

'''''Chernoff Bound'''''
:<math>
\begin{align}
\mathbf{Pr}[|X| \geq \delta] \leq \min_{t \geq 0} \frac{\mathbf{E}[e^{t|X|}]}{e^{t\delta}}
\end{align}
</math>

'''''<math>k</math>th-Moment Bound'''''
:<math>
\begin{align}
\mathbf{Pr}[|X| \geq \delta] \leq \frac{\mathbf{E}[|X|^k]}{\delta^k}
\end{align}
</math>

* Show that for each <math>\delta</math>, there exists a choice of <math>k</math> such that the <math>k</math>th-moment bound is stronger than the Chernoff bound. 
  
  '''''Hint''''': Consider the Taylor expansion of the moment generating function and apply the probabilistic method.

* Why would we still prefer the Chernoff bound to the (seemingly) stronger <math>k</math>-th moment bound?

== Problem 3 ==

Let <math> \Omega </math> be a finite set of size <math> m </math>. Let  <math>\mathcal{A}=\{A_1,A_2,\dots,A_n\}</math> be a family of <math> n </math> subsets of <math>\Omega </math>. For any "coloring" <math>\chi:\Omega\rightarrow \{-1,+1\} </math> that maps each element of <math>\Omega </math> to  <math> -1 </math> or <math> +1 </math>, and any subset <math>A\subseteq \Omega </math>, we define 
<math> 
\begin{align}
\chi(A)\triangleq \sum\limits_{a\in A}\chi(a).
\end{align}
</math>

Define the '''discrepancy''' of the family of subsets <math>\mathcal{A}</math> with respect to the "coloring"<math>\chi</math> by

<math>
\begin{align}
\quad\textsf{disc}(\mathcal{A},\chi)\triangleq \max\limits_{A\in \mathcal{A}}|\chi(A)|,
\end{align}
</math>

and the '''discrepancy''' of <math> \mathcal{A} </math> by

<math>
\begin{align}
\quad\textsf{disc}(\mathcal{A})\triangleq \min\limits_{\chi:\Omega\rightarrow\{-1,+1\}} \textsf{disc}(\mathcal{A},\chi).
\end{align}
</math>

* Show that 
<math>
\quad\textsf{disc}(\mathcal{A})\leq \sqrt{2m\ln{(2n)}}.
</math>

'''''Hint''''': Apply the probabilistic method.

* Suppose <math> m </math> is even. Show further that
<math>
\quad\textsf{disc}(\mathcal{A})\leq \sqrt{m\ln{(2n)}}.
</math>

'''''Hint''''': Set <math>\chi(i+\frac{m}{2})=-\chi(i)</math> for <math> 1\leq i\leq \frac{m}{2}</math>

== Problem 4 ==
Given an undirected graph <math>G(V, E)</math> with maximum degree <math>\Delta</math>, where the vertex set <math>V</math> is partitioned into <math>r</math> disjoint subsets, i.e., <math>V = S_1 \cup S_2 \cup \cdots \cup S_r</math> with <math>S_i \cap S_j = \varnothing</math> for all <math>i \neq j</math>. We call a vertex set <math>T</math> is a transversal of the partition <math>\{S_1, S_2, \cdots, S_r\}</math>, if <math>|T \cap S_i| = 1</math> for all <math>1 \leq i \leq r</math>. Assume that <math>|S_i| \geq 2e\Delta</math> for all <math>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>\{S_1, S_2, \cdots, S_r\}</math>.

'''Hint''': Lovász Local Lemma.

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

== Problem 5 ==
Let <math>D=(V,E)</math> be a simple directed graph with '''minimum''' outdegree <math>\delta</math> and '''maximum''' indegree <math>\Delta</math>. Prove:

For any positive integer <math> k</math>, it <math>\mathrm{e}(\Delta \delta+1)(1-\frac{1}{k})^{\delta}<1</math>, then  <math>D</math> contains a (directed,simple) cycle of length <math>0(\mod k).</math>


'''Hint''': Let <math>f:V\rightarrow \{0,1,\dots,k-1\}</math> be a random coloring of <math> V </math> obtained by choosing uniformly and independently at random for each <math> v\in V</math>. Try to use Lovász Local Lemma to show something interesting!