The Set Covering Prob.

1. The Set Covering Prob.

3. 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.

6. 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

7. 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.

8. 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

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

11. The number of elements in any SF remaining uncovered after S1, S2,…, Si have been selected by the algorithm.

13. 1/(a+1)+1/(a+2)+…1/b  1/b+1/b+…+1/b

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