Scribe : Mengfei Cao Lecture 8 : Set Cover

This lecture focused on the problem of “Set Cover”, which is known as one of the first proved 21 NP-complete problems[2]. Two formulations will be given and one optimal approximation algorithm based on a greedy strategy is introduced. Further, the problem is generalized to weighted elements and an approximation algorithm derived from an Integer Programming(IP) formulation is presented. 1 Unweighted Set Covering There are two different ways to look at the set covering problem. First we introduce the ordinary formulation, then we introduce the hitting set formulation. 1.1 Ordinary Formulation We define P , a collection of subsets of a set X as follows: X = {x1 . . . xn} (1) P = {P1 . . . Pr} (2) where xi represents a skill, Pj a person, and xi ∈ Pj if person Pj possesses the skill xi, for i = 1, ...n and j = 1, ...r. The goal is to as an employer find the minimum number of people to cover all the skills, namely to find the smallest set R ⊆ P such that the people in R covers all the skills: ∪P∈RP = X (3) An example with 6 person and 12 skills will be illustrated. In the example, by picking a set containing P1, P2, P4, P6 we could ensure that this set covers every skill. However, the set containing P3, P4, P5 has the size 3, smaller than the former set with size 4.