191014K02 | Day 4 Lecture 1

191014K02: Day 4 Lecture 1

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Isolation Lemma

Here’s our key tool for the day!

Isolation Lemma

Let $U$ be a universe with $|U|=n$ and let $\cal F$ be a family of sets over $U$. Pick a random weight function $w: U \rightarrow\{1, \cdots ,W\}$. Then:

$$\operatorname{Pr}[{\color{indianred}\cal F \text{ has a \textbf{unique} min weight set}}] \geqslant \frac{n}{W}$$

Call an element $u$ critical if:

1. $u$ is in some minimum weight set, and
2. if $w(u)$ is increased by 1 then $u$ is no longer in any minimum weight set.