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![Page 1: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/1.jpg)
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
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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
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Section 2.4-2.5
4
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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.
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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
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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’ .
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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
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Greedy Algorithm for Set-Cover
' ;
while | | ( ') do
choose to maximize ( ' { }) and
' ' { };
C
E f C
S C f C S
C C S
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Analysis
1 2
1
1
Suppose , , ..., are selected by Greedy
Algorithm. Denote { , ..., }. Then
( ) ( ) (| | ( )) /
k
i i
i i i
S S S
C S S
f C f C E f C opt
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)/11(||
Then
).(||
satisfying onelargest thebe to Choose
)/11(||
)/11))((|(|
)/11))((|(| )(||
)(|| )/11))((|(|
1
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))/|(|ln 1(
Thus,
)/||(ln So
)1:note( ||
)/11(|| /
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AnalysisAnalysis
1 2
1
1
Suppose , , ..., are selected by Greedy
Algorithm. Denote { , ..., }. Then
( ) ( ) (| | ( )) /
k
i i
i i i
S S S
C S S
f C f C E f C opt
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optCfE
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/))(||(
/))(*)((
/*))((
/))(()( Thus,
10 allfor
)()( rule,greedy By
}.,...,{* Denote
solution. optimalan be },...,,{*Let
).(}){()( Denote
10
10
1
21
1
11
11
Submodular!
Monotone!
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What’s we need?What’s we need?
)()( BfAfBA XX
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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)
)()( BfAfBA
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PropertyProperty
)(}){()( where
for )()(
ifonly and if )increasing monotone is (
for )()(
ifonly and if )submodular is (
AfxAfAf
BxBfAfBA
f
BxBfAfBA
f
x
xx
xx
1
2
![Page 18: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/18.jpg)
Proof of Proof of
)()(
Then
.\ Denote
)()()()(
)()()()(
BfBAf
BBABAABAC
BfBAfBAfAf
BAfBAfBfAf
CC
1
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)(
)()()(
))((
))(()()(
}.,...,{
denote and },...,{Let . and
allfor holds inequality above suppose ,Conversely
).()(
hence and ,},{\
Then . and }{set
, and For .submodular is Suppose
1
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21
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VBxUA
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k
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Proof of Proof of
.increasing monotone is i.e.,
),(}){(
ifonly and if
0)()(
, and For
f
AfxAf
BfAf
BxBA
xx
2
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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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optCfE
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/))(||(
/))(*)((
/*))((
/))(()( Thus,
10 allfor
)()( rule,greedy By
}.,...,{* Denote
solution. optimalan be },...,,{*Let
).(}){()( Denote
10
10
1
21
1
11
11
Why?
![Page 23: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/23.jpg)
Theorem
Greedy Algorithm produces an approximation within ln n +1 from optimal.
The same result holds for weighted set-cover.
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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’ .
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Greedy Algorithm
'.output
};{''
and )(/)'( maximize to choose
do )'(|| while
;'
C
SCC
SwCfCS
CfE
C
S
![Page 26: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/26.jpg)
Submodular Cover ProblemSubmodular Cover Problem
)( s.t.
)()( min
}.0)(,|{)( Define
.set a of subsets allon defined function
submodular ,increasing monotone aConsider
fA
xcAc
AfExAf
Ef
Ax
x
![Page 27: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/27.jpg)
Greedy AlgorithmGreedy Algorithm
))(any for )()(max*(
.output
};{
and )(/)( maximize to choose
do )(* while
;
fAAfAff
A
xAA
xcAfEx
Aff
A
x
![Page 28: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/28.jpg)
A General TheoremA General Theorem
}).({max where
ionapproximat-)ln1( a givesGreedy then ,for
0)( and function,integer an is ,0)( If
xf
Ex
xcff
Ex
)).(})({(max Therefore
).()()(
consider may we,0)( If
fxf
fAfAg
f
Ex
Remark:
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ProofProof
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and )( ,)( denote ,*
eachFor solution. optimalan is } ..., ,{* Suppose
).( and
} ..., ,{ Denote .appearance their oforder the
in AlgorithmGreedy by selected are ..., , , Suppose
1
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![Page 30: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/30.jpg)
![Page 31: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/31.jpg)
submodular is since )(
whererule,Greedy by )(
**************************
, ofproperty submodular and ruleGreedy By
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of properties increasing
monotone and submodularby )(
)*(
)(*)()()(
..., ,1for
![Page 33: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/33.jpg)
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![Page 34: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/34.jpg)
integers. are since 1
)(
)()()(
)()(
Therefore,
.
, of
property increasing monotone and submodular By the
.)(
rule,greedy By
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![Page 35: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/35.jpg)
1 2 3
![Page 36: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/36.jpg)
zekze1Ze2
![Page 37: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/37.jpg)
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.
![Page 38: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/38.jpg)
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Rank
• The rank of a graph is the maximum number of edges in an acyclic subgraph.
.submodular increasing monotone is
),( ofrank )(
r
EXGEr
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Proof
).()( Therefore,
. passing cycle acontain not does }){,(
. passing cycle acontain not does }){,(
. passing cycle acontain not does }){,(1)(
1.or 0)(}){()(
. Assume
ee BrAr
eeAX
eeBX
eeAXAr
AreArAr
BA
e
e
![Page 41: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/41.jpg)
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 ).
.submodular increasing monotone is
)()(
i
ii
r
ErEr
![Page 42: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/42.jpg)
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
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Connected Vertex-Cover
• Given a connected graph, find a minimum vertex-cover which induces a connected subgraph.
![Page 44: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/44.jpg)
• 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.
![Page 45: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062300/56649f3f5503460f94c6027d/html5/thumbnails/45.jpg)
p-q
• p-q is submodular.
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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.
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Theorem
• Connected Vertex-Cover has a 3-approximation.
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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!!!
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Thanks, End