1. Lecture 2-1 Submodular Maximization Weili Wu Ding-Zhu Du University of Texas at Dallas

2. Outline Kate Middleton effect Max Submodular Function 2

3. The trend effect that Kate, Duchess of Cambridge has on others, from cosmetic surgery for brides, to sales of coral-colored jeans.” “Kate Middleton effect Kate Middleton effect 3

4. According to Newsweek, "The Kate Effect may be worth £1 billion to the UK fashion industry." Tony DiMasso, L. K. Bennett’s US president, stated in 2012, "...when she does wear something, it always seems to go on a waiting list." Hike in Sales of Special Products 4

5. Influential persons often have many friends. Kate is one of the persons that have many friends in this social network. For more Kates, it’s not as easy as you might think! How to Find Kate? 5

6. Given a digraph and k>0, Find k seeds (Kates) to maximize the number of influenced persons (in one step). Domination Maximization 6

7. 7 Theorem Proof

8. Outline Kate Middleton effect Submodular Maximization 8

9. Max Coverage Given a collection C of subsets of a set E, find a subcollection C’ of C, with |C’|<k, to maximize the number of elements covered by C’. Influence Maximization is a special case of Max Coverage.

10. Max Coverage Given a collection C of subsets of a set E, find a subcollection C’ of C, with |C’|<k, to maximize the number of elements covered by C’.

11. Submadular Function Max 11

12. Greedy Algorithm 12

13. Performance Ratio 13 Theorem (Nemhauser et al. 1978) Proof

14. 14 Monotone increasing Why? Submodular!

15. Theorem 15

16. Exercise 16

17. Section 2.1-2.2 17

18. Independent System Consider a set E and a collection C of subsets of E. (E,C) is called an independent system if

19. Maximization c: E→R max c(A) s.t. AεC c(A) = Σ xεA c(x) +

20. Greedy Approximation MAX

21. Theorem

22. Proof

25. THANK YOU!