Multiple Products

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.

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

Greedy Algorithm

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

Proof Monotone increasing Submodular! Why?

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’ .

Knapsack Constraints

Multi-product: Knapsack Constraint

Knapsack Constraint

Knapsack Constraint

Submadular Function Max

Submadular Function Max

Naïve Greedy Algorithm Performance is not good, why?

Knapsack 1/2-approximation

An Generalization Theorem 2

Proof

Knapsack has PTAS

An Generalization Theorem 3

Proof

Matroid Constraints

Matroid constraints

Lemma 1

Greedy Approximation Theorem 1

Proof

THANK YOU!