Understanding Greedy Strategies and Submodular Functions in Set Cover Problems

1. Lecture 23 Greedy Strategy

2. What is a submodular function? Consider a function f on all subsets of a set E. f is submodular if

3. Set-Cover Given a collection C of subsets of a set E, find a minimum subcollection C’ of C such that every element of E appears in a subset in C’ .

4. Example of Submodular Function

5. Greedy Algorithm

6. Analysis

9. Analysis

11. What’s we need?

12. Actually, this inequality holds if and only if f is submodular and (monotone increasing)

18. Meaning of Submodular • The earlier, the better! • Monotone decreasing gain!

19. Theorem Greedy Algorithm produces an approximation within ln n +1 from optimal. The same result holds for weighted set-cover.

20. Weighted Set Cover Given a collection C of subsets of a set E and a weight function w on C, find a minimum total-weight subcollection C’ of C such that every element of E appears in a subset in C’ .

21. Greedy Algorithm

22. A General Problem

23. Greedy Algorithm

24. A General Theorem Remark:

25. Proof

31. 1 2 3

32. ze1 zek Ze2

33. Subset Interconnection Design • Given m subsets X1, …, Xm of set X, find a graph G with vertex set X and minimum number of edges such that for every i=1, …, m, the subgraph G[Xi] induced by Xi is connected.

34. fi For any edge set E, define fi(E) to be the number of connected components of the subgraph of (X,E), induced by Xi. • Function -fi is submodular.

35. Rank • All acyclic subgraphs form a matroid. • The rank of a subgraph is the cardinality of a maximum independent subset of edges in the subgraph. • Let Ei = {(u,v) in E | u, v in Xi}. • Rank ri(E)=ri(Ei)=|Xi|-fi(E). • Rank ri is sumodular.

36. Potential Function r1+ּּּ+rm Theorem Subset Interconnection Design has a (1+ln m)-approximation. r1(Φ)+ּּּ+rm(Φ)=0 r1(e)+ּּּ+rm(e)<m for any edge

37. Connected Vertex-Cover • Given a connected graph, find a minimum vertex-cover which induces a connected subgraph.

38. For any vertex subset A, p(A) is the number of edges not covered by A. • For any vertex subset A, q(A) is the number of connected component of the subgraph induced by A. • -p is submodular. • -q is not submodular.

39. |E|-p(A) • p(A)=|E|-p(A) is # of edges covered by A. • p(A)+p(B)-p(A U B) = # of edges covered by both A and B > p(A ∩ B)

40. -p-q • -p-q is submodular.

41. Theorem • Connected Vertex-Cover has a (1+ln Δ)-approximation. • -p(Φ)=-|E|, -q(Φ)=0. • |E|-p(x)-q(x) <Δ-1 • Δ is the maximum degree.

42. Theorem • Connected Vertex-Cover has a 3-approximation.

43. Weighted Connected Vertex-Cover Given a vertex-weighted connected graph, find a connected vertex-cover with minimum total weight. Theorem Weighted Connected Vertex-Cover has a (1+ln Δ)-approximation. This is the best-possible!!!

44. End Thanks!