The Set Covering Prob.

Application Suppose X represents a set of skills that are needed to solve a problem and that we have a given set of people available to work on the problem. We wish to form a committee, containing as few people as possible, such that for every requisite skill in X, there is a member of the committee having that skill.

How good is Greedy-Set-Cover?  (n)=H(max{|S|: S  F}) Proof: –We assign a cost of 1 to each set selected. –We distributed this cost over the elements covered for the first time. –C*: the optimal set cover –C: the set cover returned by the algorithm –Let Si be the ith subset selected

How good is Greedy-Set-Cover?  (n)=H(max{|S|: S  F}) dth harmonic number, denoted by H(d). As a boundary condition, we define H(0)=0.

Proof At each step of the algorithm, 1 unit of cost is assigned. Each element is assigned a cost only once. By the two statements above, we have

The cost assigned to the optimal set cover This is because each x  X is in at least one set S  C*

The number of elements in any S  F remaining uncovered after S 1, S 2,…, S i have been selected by the algorithm.

1/(a+1)+1/(a+2)+…1/b  1/b+1/b+…+1/b

This is because Hn = ln n + O(1)