problem setting :influence maximization a new product is available in the market. whom to give free...
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Problem Setting :Influence Maximization
• A new product is available in the market.
Whom to give free samples to maximize the purchase of the product ?
1
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
2
Section 2.4-2.5
4
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?What is a submodular function?
Consider a function f on all subsets of a set E.f is submodular if
( ) ( ) ( ) ( )f A f B f A B f A B
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
For a subcollection of , define
( ) | S|.
Then
( ) ( ) ( ) ( )
s A
A C
f A
f A f B f A B f A B
Greedy Algorithm for Set-Cover
' ;
while | | ( ') do
choose to maximize ( ' { }) and
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Analysis
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Algorithm. Denote { , ..., }. Then
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AnalysisAnalysis
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Suppose , , ..., are selected by Greedy
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Submodular!
Monotone!
What’s we need?What’s we need?
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Actually, this inequality holds if and only if Actually, this inequality holds if and only if ff is is submodular and submodular and
(monotone increasing)(monotone increasing)
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PropertyProperty
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for )()(
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for )()(
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Proof of Proof of
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Meaning of SubmodularMeaning of Submodular
• The earlier, the better!• Monotone decreasing gain!
• Submodular =discrete concave = second derivative < 0
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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
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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.
.submodular increasing monotone is
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Proof
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1.or 0)(}){()(
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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 minimumtotal weight.
Theorem Weighted Connected Vertex-Coverhas a (1+ln Δ)-approximation.
This is the best-possible!!!
Thanks, End
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