1. Multiple Products

2. Monotone Submodular Maximization
Lecture 2-1
Monotone Submodular Maximization
Weili Wu Ding-Zhu Du
University of Texas at Dallas
First, I want to thank you for you presence. ********In this presentation I will try to introduce The social network which is a theoretical structure to study relationships between individuals, groups, organizations, or even entire societies. 
It is related to a wide range of disciplines. These disciplines include, but are not limited to information science, biology, economics, geography, communication studies, and so on..
The study of social networks begins with the late eighteenth century, two sociologists (Émile [ei'mi:l] Durkheim and Ferdinand ['fɝdənænd] Fer迪南de Tönnies) foreshadowed the idea of social networks in their theories and research of social groups.
Nowadays, we study social networks using network analysis to identify social communities, pick influential person, and design good software.

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

4. What is monotone ?
f is monotone (nondecreasing) if

5. Decreasing Marginal Value1 2

6. Submadular Function Max

7. Greedy Algorithm

8. Performance Ratio
Theorem 1(Nemhauser et al. 1978)
Proof

9. Proof
Monotone increasing
Submodular!
Why?

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. Knapsack Constraints

12. Multi-product: Knapsack Constraint

13. Knapsack Constraint

14. Knapsack Constraint

15. Submadular Function Max

16. Submadular Function Max

17. Naïve Greedy Algorithm
Performance is not good, why?

18. Knapsack
1/2-approximation

19. An Generalization
Theorem 2

20. Proof

22. Knapsack has PTAS

23. An Generalization
Theorem 3

24. Proof

26. Matroid Constraints

27. Matroid constraints

28. Lemma 1

29. Greedy Approximation
Theorem 1

30. Proof

32. THANK YOU!