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.