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!