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Monotone Submodular Maximization

Lecture 2-1. Monotone Submodular Maximization. Weili Wu Ding-Zhu Du University of Texas at Dallas. Lecture 1-1. What is a submodular function?. Consider a function f on all subsets of a set E . f is submodular if. What is monotone ?. f is monotone (nondecreasing) if.

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Monotone Submodular Maximization

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

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

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

  4. Decreasing Marginal Value 1 2

  5. Submadular Function Max

  6. Greedy Algorithm

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

  8. Proof Monotone increasing Submodular! Why?

  9. Example 1: 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’ .

  10. Example 2: Influence Max • What is social network? • What is social influence? • What is influence maximization?

  11. What is Social Network? Wikipedia Definition: Social Structure • Nodes:Social actors (individuals or organizations) • Links:Social relations

  12. What is Social Influence? [1] http://en.wikipedia.org/wiki/Social_influence • Social influence occurs when one's opinions, emotions, or behaviors are affected by others, intentionally or unintentionally.[1] • Informational social influence: to accept information from another; • Normative social influence: to conform to the positive expectations of others.

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

  14. Hike in Sales of Special Products • 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."

  15. How to Find Kate? • 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!

  16. Influence Maximization • Given a digraph and k>0, • Find k seeds (Kates) to maximize the number of influenced persons (possibly in many steps).

  17. Theorem Proof

  18. Modularity of Influence

  19. Theorem

  20. Knapsack Constraints

  21. Multi-product: Knapsack Constraint

  22. Knapsack Constraint

  23. Knapsack Constraint

  24. Submadular Function Max

  25. Submadular Function Max

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

  27. Knapsack 1/2-approximation

  28. An Generalization Theorem 2

  29. A Variation Theorem 2

  30. Proof

  31. Knapsack has PTAS Why?

  32. Why?

  33. Performance Analysis

  34. Time

  35. An Generalization Theorem 3

  36. Proof

  37. Matroid Constraints

  38. Matroid constraints

  39. Lemma 1

  40. Greedy Approximation Theorem 1

  41. Proof

  42. THANK YOU!

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