随机算法 (Spring 2014)/Problem Set 2

Problem 1

(Due to D.R. Karger and R. Motwani.)

1. Let [math]\displaystyle{ S,T }[/math] be two disjoint subsets of a universe [math]\displaystyle{ U }[/math] such that [math]\displaystyle{ |S|=|T|=n }[/math]. Suppose we select a random set [math]\displaystyle{ R\subseteq U }[/math] by independently sampling each element of [math]\displaystyle{ U }[/math] with probability [math]\displaystyle{ p }[/math]. We say that the random sample [math]\displaystyle{ R }[/math] is good if the following two conditions hold: [math]\displaystyle{ R\cap S=\emptyset }[/math] and [math]\displaystyle{ R\cap T\ne\emptyset }[/math]. Show that for [math]\displaystyle{ p=1/n }[/math], the probability that [math]\displaystyle{ R }[/math] is good is larger than some positive constant.
2. Suppose now that the random set [math]\displaystyle{ R }[/math] is chosen by sampling the elements of [math]\displaystyle{ U }[/math] with only pairwise independence. Show that for a suitable choice of the value of [math]\displaystyle{ p }[/math], the probability that [math]\displaystyle{ R }[/math] is good is larger than some positive constant.