Seed Selection

Problem Setting :Influence Maximization A new product is available in the market. Whom to give free samples to maximize the purchase of the product ? To study the effects of “word of mouth” in the promotion of new products, Domingos and Richardson posed an algorithmic problem in the year 2001.

Problem Setting :Seed Selection A new product is available in the market. Whom to give free samples to minimize the budget in order to occupy certain market ? To study the effects of “word of mouth” in the promotion of new products, Domingos and Richardson posed an algorithmic problem in the year 2001.

Problem Setting: Min Seeding Given a market (e.g. a set of individuals) estimates for influence between individuals Goal Minimum budget for initial advertising (e.g. give away free samples of product) in order to occupy the market. Question Which set of individuals should we target at? Application besides product marketing spread an innovation, ideas, news detect stories in blogs analyze Twitter if we can try to convince a subset of individuals to adopt a new product or innovation But how should we choose the few key individuals to use for seeding this process? Which blogs should one read to be most up to date?

Outline Minimum submodular cover Seed selection

Part I Min Submodular Cover Weili Wu Ding-Zhu Du

Section 2.4-2.5

Max and Min Min f is equivalent to Max –f. However, a good approximation for Min f may not be a good approximation for Min –f. For example, consider a graph G=(V,E). C is a minimum vertex cover of G if and only if V-C is a maximum independent of G. The minimum vertex cover has a polynomial-time 2-approximation, but the maximum independent set has no constant-bounded approximation unless NP=P.

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

Min Set-Cover Given a collection C of subsets of a set E, find a minimum subcollection C’ of C such that every element of E appears in a subset in C’ .

Example of Submodular Function

Greedy Algorithm for Set-Cover

Analysis

Analysis

Submodular! Monotone!

What’s we need?

Actually, this inequality holds if and only if f is submodular and (monotone increasing)

Property 1 2

Proof of 1

Proof of 2

Meaning of Submodular The earlier, the better! Monotone decreasing gain! Submodular =discrete concave = second derivative < 0

Why?

Theorem Greedy Algorithm produces an approximation within ln n +1 from optimal. The same result holds for weighted set-cover.

Weighted Set Cover Given a collection C of subsets of a set E and a weight function w on C, find a minimum total-weight subcollection C’ of C such that every element of E appears in a subset in C’ .

Greedy Algorithm

Submodular Cover Problem

Greedy Algorithm

A General Theorem Remark:

Proof

1 2 3

ze1 zek Ze2

Subset Interconnection Design Given m subsets X1, …, Xm of set X, find a graph G with vertex set X and minimum number of edges such that for every i=1, …, m, the subgraph G[Xi] induced by Xi is connected.

Rank The rank of a graph is the maximum number of edges in an acyclic subgraph.

Proof

Rank The rank of a graph is the maximum of edgces in an acyclic subgraph. Let Ei = {(u,v) in E | u, v in Xi}. Gi =(X,Ei ).

Potential Function r1+ּּּ+rm Theorem Subset Interconnection Design has a (1+ln m)-approximation. r1(Φ)+ּּּ+rm(Φ)=0 r1(e)+ּּּ+rm(e)<m for any edge

Connected Vertex-Cover Given a connected graph, find a minimum vertex-cover which induces a connected subgraph.

For any vertex subset A, p(A) is the number of edges covered by A. For any vertex subset A, q(A) is the number of connected component of the subgraph induced by A. p is monotone increasing submodular. -q is not submodular.

p-q p-q is submodular.

Theorem Connected Vertex-Cover has a (1+ln Δ)-approximation. p(Φ)=0, -q(Φ)=0. p(x)-q(x) < Δ-1 Δ is the maximum degree.

Theorem Connected Vertex-Cover has a 3-approximation.

Weighted Connected Vertex-Cover Given a vertex-weighted connected graph, find a connected vertex-cover with minimum total weight. Theorem Weighted Connected Vertex-Cover has a (1+ln Δ)-approximation. This is the best-possible!!!

Seed Selection

Min Seeding

Greedy Algorithm

Greedy Algorithm

Theorem

Thanks, End