随机算法 (Fall 2011)/Problem set 3

Problem 3

The set cover problem is defined as follows:

• Let [math]\displaystyle{ U=\{u_1,u_2,\ldots,u_n\} }[/math] be a set of [math]\displaystyle{ n }[/math] elements, and let [math]\displaystyle{ \mathcal{S}=\{S_1,S_2,\ldots,S_m\} }[/math] be a family of subsets of [math]\displaystyle{ U }[/math]. For each [math]\displaystyle{ u_i\in U }[/math], let [math]\displaystyle{ w_i }[/math] be a nonnegative weight of [math]\displaystyle{ u_i }[/math]. The goal is to find a subset [math]\displaystyle{ V\subseteq U }[/math] with the minimum total weight [math]\displaystyle{ \sum_{i\in V}w_i }[/math], that intersects with all [math]\displaystyle{ S_i\in\mathcal{S} }[/math].

This problem is NP-hard.

(Remark: There are two equivalent definitions of the set cover problem. We take the hitting set version.)

Questions:

1. Give an integer programming for the problem and its linear programming relaxation.
2. Consider the following idea of randomized rounding: independently round each fractional value to [math]\displaystyle{ \{0,1\} }[/math]; and repeatedly apply this process to the variables rounded to 0 until every subset [math]\displaystyle{ S_i\in\mathcal{S} }[/math] is "hited" once.

Use this idea to develop an randomized approximation algorithm and prove that your algorithm returns an [math]\displaystyle{ O(\log m) }[/math]-approximate solution with probability at least [math]\displaystyle{ \frac{1}{2} }[/math].