4.1 Set Cover

Theorem 4.1.1 If OPT contains k sets, the greedy algorithm uses ≤ k(1 + ln nk ) sets. Proof: Let It be the sets selected by the greedy algorithm up to t iterations. Let nt be the number of uncovered elements at iteration t. Then nt = n− | ⋃ i∈It Si|, n0 = n, I0 = ∅. We claim that: Claim 4.1.2 nt ≤ (1− 1 k )nt−1 Proof: Let Jt = U\( ⋃ i∈It Si), then OPT covers Jt−1 with ≤ k sets. Because |Jt−1| = nt−1, we know that OPT covers nt−1 uncovered elements with ≤ k sets. Therefore there exists a set in OPT which covers at least nt−1 k uncovered elements. Because the greedy algorithm always chooses the set which covers most uncovered elements, the greedy algorithm covers at least nt−1 k uncovered elements at iteration t. Therefore nt ≤ nt−1 − nt−1 k = (1− 1 k )nt−1 Now, by induction, nt ≤ (1− 1 k ) tn.