1. Problem Setting :Influence Maximization A new product is available in the market. Whom to give free samples to maximize the purchase of the product ? 1

2. Problem Setting: Min Seeding Given – a market (e.g. a set of individuals) – estimates for influence between individuals Goal – Minimum budget for initial advertising (e.g. give away free samples of product) in order to occupy the market. Question – Which set of individuals should we target at? Application besides product marketing – spread an innovation, ideas, news – detect stories in blogs – analyze Twitter 2

3. Lecture 2-1 Min Submodular Cover Weili Wu Ding-Zhu Du

4. Section 2.4-2.5 4

5. Max and Min Min f is equivalent to Max –f. However, a good approximation for Min f may not be a good approximation for Min –f. For example, consider a graph G=(V,E). C is a minimum vertex cover of G if and only if V-C is a maximum independent of G. The minimum vertex cover has a polynomial-time 2-approximation, but the maximum independent set has no constant- bounded approximation unless NP=P.

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

7. Min 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’.

8. Example of Submodular Function

9. Greedy Algorithm for Set-Cover

10. Analysis

13. Analysis

14. Submodular! Monotone!

15. What’s we need?

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

17. Property 1 2

18. Proof of 1

20. 2

21. Meaning of Submodular The earlier, the better! Monotone decreasing gain! Submodular =discrete concave = second derivative < 0

22. Why?

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

24. 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’.

25. Greedy Algorithm

26. Submodular Cover Problem

27. Greedy Algorithm

28. A General Theorem Remark:

29. Proof

35. 12 3

36. z ek z e1 Ze2Ze2

37. Subset Interconnection Design Given m subsets X 1, …, X m 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[X i ] induced by X i is connected.

39. Rank The rank of a graph is the maximum number of edges in an acyclic subgraph.

40. Proof

41. Rank The rank of a graph is the maximum of edgces in an acyclic subgraph. Let E i = {(u,v) in E | u, v in X i }. G i =(X,E i ).

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

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

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

45. p-q p-q is submodular.

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

47. Theorem Connected Vertex-Cover has a 3- approximation.

48. 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!!!

49. Thanks, End