problem setting :influence maximization a new product is available in the market. whom to give free...
TRANSCRIPT
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
' ' { };
C
E f C
S C f C S
C C S
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
i
i
i
i
ii
ii
optEopt
optik
CfEopt
i
optE
optCfE
optCfECfE
CfEoptCfE
)/11(||
Then
).(||
satisfying onelargest thebe to Choose
)/11(||
)/11))((|(|
)/11))((|(| )(||
)(|| )/11))((|(|
1
21
1
1
))/|(|ln 1(
Thus,
)/||(ln So
)1:note( ||
)/11(|| /
optEopt
ioptk
optEopti
exeE
optEoptxopti
i
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
optCfE
optCfCCf
optCCf
optCfCf
optj
CfCf
XXC
XXXC
AfXAfAf
i
ii
optjjiX
optjiXiS
iXiS
jj
opt
X
j
ji
ji
/))(||(
/))(*)((
/*))((
/))(()( Thus,
10 allfor
)()( rule,greedy By
}.,...,{* Denote
solution. optimalan be },...,,{*Let
).(}){()( Denote
10
10
1
21
1
11
11
Submodular!
Monotone!
What’s we need?What’s we need?
)()( BfAfBA XX
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
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
Proof of Proof of
)()(
Then
.\ Denote
)()()()(
)()()()(
BfBAf
BBABAABAC
BfBAfBAfAf
BAfBAfBfAf
CC
1
)(
)()()(
))((
))(()()(
}.,...,{
denote and },...,{Let . and
allfor holds inequality above suppose ,Conversely
).()(
hence and ,},{\
Then . and }{set
, and For .submodular is Suppose
1
1
1
1
1
1
21
21
ky
yyC
ky
yyC
ii
k
xx
CBf
CBfBfBf
CBAf
CBAfBAfBAf
yyC
yyCVxVU
VfUf
UBAxBA
VBxUA
VxVUf
k
k
Proof of Proof of
.increasing monotone is i.e.,
),(}){(
ifonly and if
0)()(
, and For
f
AfxAf
BfAf
BxBA
xx
2
Meaning of SubmodularMeaning of Submodular
• The earlier, the better!• Monotone decreasing gain!
• Submodular =discrete concave = second derivative < 0
optCfE
optCfCCf
optCCf
optCfCf
optj
CfCf
XXC
XXXC
AfXAfAf
i
ii
optjjiX
optjiXiS
iXiS
jj
opt
X
j
ji
ji
/))(||(
/))(*)((
/*))((
/))(()( Thus,
10 allfor
)()( rule,greedy By
}.,...,{* Denote
solution. optimalan be },...,,{*Let
).(}){()( Denote
10
10
1
21
1
11
11
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
'.output
};{''
and )(/)'( maximize to choose
do )'(|| while
;'
C
SCC
SwCfCS
CfE
C
S
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
Greedy AlgorithmGreedy Algorithm
))(any for )()(max*(
.output
};{
and )(/)( maximize to choose
do )(* while
;
fAAfAff
A
xAA
xcAfEx
Aff
A
x
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:
ProofProof
eii
ik
i i
ie
k
i k
kek
i
iieeie
iiiyeie
h
ixi
ii
k
zr
c
r
cz
r
c
r
cz
r
czzyw
xccAfzAy
yyA
Afr
xxA
xxx
e
i
)(
)()(
and )( ,)( denote ,*
eachFor solution. optimalan is } ..., ,{* Suppose
).( and
} ..., ,{ Denote .appearance their oforder the
in AlgorithmGreedy by selected are ..., , , Suppose
1
1
2
11
1
1
1
1,
1
1
1
1
21
submodular is since )(
whererule,Greedy by )(
**************************
, ofproperty submodular and ruleGreedy By
2
2
2
1
02
0
1
1
2
2
1
1
2
2
fc
r
c
Af
Ac
Af
c
r
c
r
c
r
c
r
f
x
x
k
k
h
j
ji
h
j
iy
h
j
jiy
k
ij
iiikj
k
ij Ay
eij
z
f
Af
AAf
AfAAfAfAfr
kizr
j
j
e
1
1
1
1
11
111
*
of properties increasing
monotone and submodularby )(
)*(
)(*)()()(
..., ,1for
*
)(
)(
)(
)(
)(
*1
1
2*
11
1
1
1
211
1
1
1
1
1
Ae
ee
y
e
Ay
eii
ik
i i
i
Ay
e
k
ij
ji
ik
i i
ik
j
j
k
kj
j
k
ij
k
ij k
kjj
k
i i
i
i
k
i i
ik
yw
zr
c
r
cz
r
c
rr
c
r
cr
r
c
rr
crr
r
c
cr
rAc
integers. are since 1
)(
)()()(
)()(
Therefore,
.
, of
property increasing monotone and submodular By the
.)(
rule,greedy By
1
1
1
1
1,
1
1
1,
1,
ei
z
i
e
ek
eek
k
i ei
eieei
k
kek
k
i i
iieeie
ieei
ei
e
i
i
zi
yc
z
ycz
z
yczz
r
cz
r
czzyw
zz
f
z
yc
r
c
e
1 2 3
zekze1Ze2
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
),( ofrank )(
r
EXGEr
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
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
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