Randomized Algorithms (Spring 2010)/The probabilistic method: Difference between revisions
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:If <math>{n\choose k}\cdot 2^{1-{k\choose 2}}<1</math> then it is possible to color the edges of <math>K_n</math> with two colors so that there is no monochromatic <math>K_k</math> subgraph. | :If <math>{n\choose k}\cdot 2^{1-{k\choose 2}}<1</math> then it is possible to color the edges of <math>K_n</math> with two colors so that there is no monochromatic <math>K_k</math> subgraph. | ||
|} | |} | ||
'''Proof:''' Consider a random two-coloring of edges of <math>K_n</math> | '''Proof:''' | ||
Consider a random two-coloring of edges of <math>K_n</math> obtained as follows: | |||
* For each edge of <math>K_n</math>, independently flip a fair coin to decide the color of the edge. | |||
For any fixed set <math>S</math> of <math>k</math> vertices, let <math>\mathcal{E}_S</math> be the event that the <math>K_k</math> subgraph induced by <math>S</math> is monochromatic. There are <math>{k\choose 2}</math> many edges in <math>K_k</math>, therefore | |||
:<math>\Pr[\mathcal{E}_S]=2\cdot 2^{-{k\choose 2}}=2^{1-{k\choose 2}}.</math> | |||
Since there are <math>{n\choose k}</math> possible choices of <math>S</math>, by the union bound | |||
:<math> | |||
\Pr[\exists S, \mathcal{E}_S]\le {n\choose k}\cdot\Pr[\mathcal{E}_S]={n\choose k}\cdot 2^{1-{k\choose 2}}. | |||
</math> | |||
Due to the assumption, <math>{n\choose k}\cdot 2^{1-{k\choose 2}}<1</math>, thus there exists a two coloring that none of <math>\mathcal{E}_S</math> occurs, which means there is no monochromatic <math>K_k</math> subgraph. | |||
<math>\square</math> | |||
Note that for <math>k\ge 3</math> and we take <math>n=\lfloor2^{k/2}\rfloor</math>, then | |||
:<math> | |||
\begin{align} | |||
{n\choose k}\cdot 2^{1-{k\choose 2}} | |||
&< | |||
\frac{n^k}{k!}\cdot\frac{2^{1+\frac{k}{2}}}{2^{k^2/2}}\\ | |||
&\le | |||
\frac{2^{k^2/2}}{k!}\cdot\frac{2^{1+\frac{k}{2}}}{2^{k^2/2}}\\ | |||
&= | |||
\frac{2^{1+\frac{k}{2}}}{k!}\\ | |||
&<1. | |||
\end{align} | |||
</math> | |||
;Circuit complexity | ;Circuit complexity |
Revision as of 05:57, 17 April 2010
The Basic Idea
Counting or sampling
- Ramsey number
Theorem
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Proof: Consider a random two-coloring of edges of [math]\displaystyle{ K_n }[/math] obtained as follows:
- For each edge of [math]\displaystyle{ K_n }[/math], independently flip a fair coin to decide the color of the edge.
For any fixed set [math]\displaystyle{ S }[/math] of [math]\displaystyle{ k }[/math] vertices, let [math]\displaystyle{ \mathcal{E}_S }[/math] be the event that the [math]\displaystyle{ K_k }[/math] subgraph induced by [math]\displaystyle{ S }[/math] is monochromatic. There are [math]\displaystyle{ {k\choose 2} }[/math] many edges in [math]\displaystyle{ K_k }[/math], therefore
- [math]\displaystyle{ \Pr[\mathcal{E}_S]=2\cdot 2^{-{k\choose 2}}=2^{1-{k\choose 2}}. }[/math]
Since there are [math]\displaystyle{ {n\choose k} }[/math] possible choices of [math]\displaystyle{ S }[/math], by the union bound
- [math]\displaystyle{ \Pr[\exists S, \mathcal{E}_S]\le {n\choose k}\cdot\Pr[\mathcal{E}_S]={n\choose k}\cdot 2^{1-{k\choose 2}}. }[/math]
Due to the assumption, [math]\displaystyle{ {n\choose k}\cdot 2^{1-{k\choose 2}}\lt 1 }[/math], thus there exists a two coloring that none of [math]\displaystyle{ \mathcal{E}_S }[/math] occurs, which means there is no monochromatic [math]\displaystyle{ K_k }[/math] subgraph.
[math]\displaystyle{ \square }[/math]
Note that for [math]\displaystyle{ k\ge 3 }[/math] and we take [math]\displaystyle{ n=\lfloor2^{k/2}\rfloor }[/math], then
- [math]\displaystyle{ \begin{align} {n\choose k}\cdot 2^{1-{k\choose 2}} &\lt \frac{n^k}{k!}\cdot\frac{2^{1+\frac{k}{2}}}{2^{k^2/2}}\\ &\le \frac{2^{k^2/2}}{k!}\cdot\frac{2^{1+\frac{k}{2}}}{2^{k^2/2}}\\ &= \frac{2^{1+\frac{k}{2}}}{k!}\\ &\lt 1. \end{align} }[/math]
- Circuit complexity
Theorem (Shannon)
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- Blocking number
Let [math]\displaystyle{ S }[/math] be a set. Let [math]\displaystyle{ 2^{S}=\{A\mid A\subseteq S\} }[/math] be the power set of [math]\displaystyle{ S }[/math], and let [math]\displaystyle{ {S\choose k}=\{A\mid A\subseteq S\mbox{ and }|A|=k\} }[/math] be the [math]\displaystyle{ k }[/math]-uniform of [math]\displaystyle{ S }[/math].
We call [math]\displaystyle{ \mathcal{F} }[/math] a set family (or a set system) with ground set [math]\displaystyle{ S }[/math] if [math]\displaystyle{ \mathcal{F}\subseteq 2^{S} }[/math]. The members of [math]\displaystyle{ \mathcal{F} }[/math] are subsets of [math]\displaystyle{ S }[/math].
Given a set family [math]\displaystyle{ \mathcal{F} }[/math] with ground set [math]\displaystyle{ S }[/math], a set [math]\displaystyle{ T\subseteq S }[/math] is a blocking set of [math]\displaystyle{ \mathcal{F} }[/math] if all [math]\displaystyle{ A\in\mathcal{F} }[/math] have [math]\displaystyle{ A\cap T\neq \emptyset }[/math], i.e. [math]\displaystyle{ T }[/math] intersects (blocks) all member set of [math]\displaystyle{ \mathcal{F} }[/math].
Theorem
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Proof: Let [math]\displaystyle{ \tau=\left\lceil\frac{n\ln m}{k}\right\rceil }[/math]. Let [math]\displaystyle{ T }[/math] be a set chosen uniformly at random from [math]\displaystyle{ {S\choose \tau} }[/math]. We show that [math]\displaystyle{ T }[/math] is a blocking set of [math]\displaystyle{ \mathcal{F} }[/math] with a probability >0.
Fix any [math]\displaystyle{ A\in\mathcal{F} }[/math]. Recall that [math]\displaystyle{ \mathcal{F}\subseteq{S\choose k} }[/math], thus [math]\displaystyle{ |A|=k }[/math]. And
- [math]\displaystyle{ \begin{align} \Pr[A\cap T=\emptyset] &= \frac{\left|{S-T\choose \tau}\right|}{\left|{S\choose \tau}\right|}\\ &= \frac{{n-k\choose \tau}}{{n\choose\tau}}\\ &= \frac{(n-k)\cdot(n-k-1)\cdots(n-k-\tau+1)}{n\cdot(n-1)\cdots(n-\tau+1)}\\ &\lt \left(1-\frac{k}{n}\right)^{\tau}\\ &\le \exp\left(-\frac{k\tau}{n}\right)\\ &\le \frac{1}{m}. \end{align} }[/math]
By the union bound, the probability that there exists an [math]\displaystyle{ A\in\mathcal{F} }[/math] that misses [math]\displaystyle{ T }[/math]
- [math]\displaystyle{ \Pr[\exists A\in\mathcal{F}, A\cap T=\emptyset]\le m\Pr[A\cap T=\emptyset]\lt m\cdot\frac{1}{m}=1. }[/math]
Thus, the probability that [math]\displaystyle{ T }[/math] is a blocking set
- [math]\displaystyle{ \Pr[\forall A\in\mathcal{F}, A\cap T\neq\emptyset]\gt 0. }[/math]
There exists a blocking set of size [math]\displaystyle{ \tau=\left\lceil\frac{n\ln m}{k}\right\rceil }[/math].
[math]\displaystyle{ \square }[/math]
Linearity of expectation
Theorem
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Theorem
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Alterations
- Independent sets
Theorem
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- Dominating sets
Theorem
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- Intersecting families
The Erdős-Ko-Rado theorem
Theorem (Erdős-Ko-Rado 1961)
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Proof (due to Katona 1972).
- Two-coloring of hypergraphs
Theorem (Erdős 1963)
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Theorem (Radhakrishnan and Srinivasan 2000)
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