Set Cover

1. Set Cover 資工碩一9562635簡裕峰

2. Set Cover • Problem 2.1 (Set Cover) Given a universe U of n elements, a collection of subsets of U, S ={S1,…,Sk}, and a cost function c : S -> Q+, find a minimum cost subcollection of S that covers all elements of U.

3. Set Cover - ex • Weighted • U = {a, b, c, d, e} • S = {S1, S2, S3, S4} • S1 = {a, b, c} , c(S1)=5 • S2 = {b, c, d} , c(S2)=2 • S3 = {d, e} , c(S3)=3 • S4 = {a, c} , c(S4)=2 • Unit • U = {a, b, c, d, e} • S = {S1, S2, S3, S4} • S1 = {a, b, c} • S2 = {b, c, d} • S3 = {d, e} • S4 = {a, c}

4. Set Cover • Define the frequency of an element to be the number of sets it is in. Let us denote the frequency of the most frequent element by f. • The various approximation algorithms for set cover achieve one of two factors : • O(log n) • f

5. S1 b a S2 S5 d c S4 S3 e Set Cover & Vertex Cover • When f = 2 : (Ex2.7) • U = {a, b, c, d, e} • S1 = {a, b} • S2 = {a} • S3 = {d, e} • S4 = {c, e} • S5 = {b, c, d} • Factor 2 approximation algorithm in Chapter 1

6. Set Cover & Vertex Cover • When f = 2 : (Ex2.7) • U = {a, b, c, d, e, f} • S1 = {a, b} • S2 = {a, f} • S3 = {d, e} • S4 = {c, e} • S5 = {b, c, d} • Factor 2 approximation algorithm in Chapter 1 S1 b a S2 S5 d f c S4 S3 e

7. 2.1 The Greedy algorithm • Algorithm 2.2 Greedy set cover algorithm Iteratively pick the most cost-effective set and remove the covered elements, until all elements are covered.

8. 2.1 The Greedy algorithm • Weighted • U = {a, b, c, d, e} • S = {S1, S2, S3, S4} • S1 = {a, b, c} , c(S1)=5 • S2 = {b, c, d} , c(S2)=2 • S3 = {d, e} , c(S3)=3 • S4 = {a, c} , c(S4)=2 • Unit • U = {a, b, c, d, e} • S = {S1, S2, S3, S4} • S1 = {a, b, c} • S2 = {b, c, d} • S3 = {d, e} • S4 = {a, c}

9. 2.1 The Greedy algorithm • Let e1,…,en be this numbering. • Lemma 2.3 • Prove

10. 2.1 The Greedy algorithm • Theorem 2.4 The greedy algorithm is an Hn factor approximation algorithm for the minimun set cover problem, where Hn = 1 + ½ + … + 1/n. Prove : The total cost = By lemma 2.3, this is at most Hn xOPT

11. 2.1 The Greedy algorithm • Example 2.5 The following is a tight example for algorithm 2.2 … 1+ε 1/n 1/(n-1) 1

12. 2.2 Layering • Let w : V -> Q+ be the function assigning weights to the vertices of the given graph G = (V,E). • Degree-weighted: if there is a constant c > 0 such that the weight of each vertex v V is c x deg(v).

13. S1 b a S2 S5 d c S4 S3 e 2.2 Layering • Degree-weighted • w(S1) = 2c • w(S2) = 1c • w(S3) = 2c • w(S4) = 2c • w(S5) = 3c

14. 2.2 Layering • Lemma 2.6 Let w : V -> Q+ be a degree-weighted function. Then w(V) ≤ 2OPT. Prove OPT ≥ |E|, w(V) ≤ 2|E|

15. 2.2 Layering • Let us define the largest degree-weighted function in w as follow: • Remove all degree zero vertices from the graph • Over the remaining vertices, compute c = min{ w(v) / deg(v)}, • t(v) = c x deg(v) is the desired function • w`(v) = w(v) – t(v) to be the residual weight function.

16. 2.2 Layering Gk Dk Gk-1 Wk-1 Dk-1 G1 W1 D1 G0 W0 D0

17. 7 3 4 3 0 3 5 3 2 1 3 4 3 4 7 1 3 2 2.2 Layering

18. 4 2 2 3 1 2 3 0 3 2 1 1 2.2 Layering 3

19. 2.2 Layering • 7 3 5 4 7 3 • 4 ~ 3 3 3 2 • 2 ~ 1 x ~ 1

20. 2.2 Layering • Theorem 2.7 The layer algorithm achieves an approximation guarantee of factor 2 for the vertex cover problem, assuming arbitrary vertex weights. Prove 1. ? The vertices we chosen is a vertex cover.

21. 2.2 Layering • Prove 2.8 Let C* be an optimal vertex cover w(C) ≤ 2 OPT If v Wj , w(v) = If v V – C, w(v) ≥

22. 2.3 Application to shortest superstring • Problem 2.9 Given a finite alphabet ∑, and a set of n strings, S = {s1,…,sn} ∑+, find a shortest string s that contains each sias a substring. Without loss of generality, we may assume that no string si is a substring of another string sj, i≠j.

23. 2.3 Application to shortest superstring • Algorithm 2.10 1. Use the greedy set cover algorithm to find a cover for the instance S. Let set( ),…, set( ) be the sets picked by this cover. 2.Concatenate the string ,…, , in any order. 3.Output the resulting string. Say s.

24. 2.3 Application to shortest superstring • Lemma 2.11 OPT ≤ OPTs ≤ 2OPT • OPT : the length of the shortest superstring • OPTs : an optimal solution to S

25. 2.3 Application to shortest superstring

26. 2.3 Application to shortest superstring • Theorem 2.12 This algorithm is a 2Hn factor algorithm for the shortest superstring problem, where n is the number of strings in the given instance.