Lecture 2 1 Monotone Submodular Maximization Weili Wu

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

Decreasing Marginal Value 1 2

Submadular Function Max 5

Greedy Algorithm 6

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

Proof Monotone increasing Submodular! Why? 8

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’.

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

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

What is 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. [1] http: //en. wikipedia. org/wiki/Social_influence 12

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. ” 13

Hike in Sales of Special Products n According to Newsweek, "The Kate Effect may be worth £ 1 billion to the UK fashion industry. " n Tony Di. Masso, L. K. Bennett’s US president, stated in 2012, ". . . when she does wear something, it always seems to go on a waiting list. " 14

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! 15

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

Theorem Proof 17

Modularity of Influence 18

Theorem 19

Knapsack Constraints

Multi-product: Knapsack Constraint

Knapsack Constraint

Knapsack Constraint

Submadular Function Max 24

Submadular Function Max 25

Naïve Greedy Algorithm Performance is not good, why? 26

Knapsack 1/2 -approximation

An Generalization Theorem 2 28

A Variation Theorem 2 29

Proof 30

31

Knapsack has PTAS Why?

Why?

Performance Analysis

Time

An Generalization Theorem 3 36

Proof 37

38

Matroid Constraints

Matroid constraints

Lemma 1

Greedy Approximation Theorem 1 42

Proof

THANK YOU!