Threshold Influence Model for Allocating Advertising Budgets · Threshold Influence Model for...
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Threshold Influence Modelfor Allocating Advertising Budgets
Nao KakimuraUniversity of Tokyo
Shonan 2016 2016/04/12
Joint work with Takuro Fukunaga, Atsushi Miyauchi, Yuni Iwamasa
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Modelling information diffusion
Simple algorithm for optimization
Topic: Budget Allocation for Marketing2
TV Newspaper DM Social ads
…
Target customers
Q. How can we allocate budgetsfor effective advertisement?
Mediasources
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Simple Model: Maximum Coverage3
Maximize #customers watching ads max. f (X)
s.t. |X|≦k}1|)(|:{#)( XtNtXf
TV newspaper DM Socail ads
TV and DM → 5 people are covered
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Two Influence Models[Alon et al.12]4
Source-side model
Each source has a probability
Target-side model
Each target has a probability
max.
s.t. |X|≦k
XtNs
spt)(
11]influencedisPr[
|)(|11]influencedisPr[
XtN
tpt
Tt
tXf ]Pr[)( Expected #influenced customers
≈ max. thresholds coverage
⊆ max. monotone submodular fns
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Maximum Thresholds Coverage5
Each customer has a threshold max. f (X)
s.t. |X|≦k}|)(|:{#)( tXtNtXf
TV newspaperDM Social ads
1 10 2 6 3 3 1 2
1 2 2 2 1 0 0 0 Received influence
influenced iffsum ≧ threshold
Threshold θt
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Maximum Thresholds Coverage6
Observation
Each threshold is 1 = maximum coverage
(1-1/e)-approximation
Harder than the densest k-subgraph problem
f is not submodular or supermodular in general
If θ is randomly chosen, (1-1/e)-approximation
→ Today: Different objective on the same model
max. f (X)
s.t. |X|≦k#customers whose
received influence ≧ θt
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This Talk: Problem with Different Objective7
Maximizing cost-effectiveness
performance of budgets without constraint
NP-hard
maxf (X)
| X |
# customers influenced
the cost spent
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Result: Cost-Effectiveness Approximation8
1) Decremental greedy algorithm
Repeatedly remove one source w./ min. contribution
Approx. factor = max deg(t)
Works for a more general case
when influence is submodular
when sources/targets have weights
2) LP-based algorithm
Approx. factor = max (deg(t) – θt +1)
T
S
deg(t)
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Result: Cost-Effectiveness Approximation9
1) Decremental greedy algorithm
Repeatedly remove one source w./ min. contribution
Approx. factor = max deg(t)
Works for a more general case
when influence is submodular
when sources/targets have weights
2) LP-based algorithm
Approx. factor = max (deg(t) – θt +1)
T
S
deg(t)
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LP for Threshold Model10
Generalizing LP for densest subgraph [Charikar00]
Each source s : xs
Each target t : yt
max yttÎT
å
sub to Covering constraints
At least θt of neighboring xs’s are 1
⇔ yt =1
tt
ts
s yx ofneighbor:
It is not good to have only
(θt =deg(t)=2 for any t)
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Constraint choosing 2 out of 311
At least 2 of xs’s are 1 ⇔ yt =1
⇔ Sum of the 2 smallest ones is ≧1
x1 + x2 ³ ytx2 + x3 ³ yt
x1 + x3 ³ yt
θt=2
x1
x2
x3
yt
011 001 000 111
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Constraint choosing θt out of d=deg(t)12
At least θt of xs’s are 1 ⇔ yt =1
⇔ Sum of the pt smallest ones is ≧1
θt
x1
x2
xd
yt
0011
pt = deg(t) – θt +1
θt
for any choice of pt-size subsets…
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LP for Threshold Model13
Rem. This LP can be solved in poly-time
target t is valid if the sum of minimum pt values ≧ yt
max yttÎT
å
xssÎU
å ³ ytTt
xssÎS
å =1
Exponential #consts
tpUtNU ||),(
← Normalization
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LP for Threshold Model14
max yttÎT
å
xssÎU
å ³ ytTt
xssÎS
å =1
Exponential #consts
tpUtNU ||),(
← Normalization
[Thm]
LP-optimal value ≧ optimal cost-effectiveness
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Rounding Algorithm15
Find an LP-optimal solution x, y
For a number r in [0,1], define X(r) = { s | xs≧ r/p }
Return the best X(r) by changing r
tTt
pp
max
r/p
x
X(r)
[Thm]
Rounded solution X(r) is p-approximation
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Rounding Algorithm16
Find an LP-optimal solution x, y
For a number r in [0,1], define X(r) = { s | xs≧ r/p }
Return the best X(r) by changing r
tTt
pp
max
r/p
x
X(r)
Covering at least
Y(r) = { t | yt≧ r }
Sum of the pt smallest ones ≧ r
[lemma]
p
LPOPT
rX
rY
|)(|
|)(|r
⇔ The θt th smallest ones ≧ r/p
⇔ θt of xs’s are in X(r)
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Proof Idea of Lemma17
If every r satisfies |)(||)(| rXp
LPrY
1
0
1
0|)(||)(| drrX
p
LPdrrY
max yttÎT
å
xssÎU
å ³ yt
xssÎS
å =1
[lemma]
p
LPOPT
rX
rY
|)(|
|)(|r
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Proof Idea of Lemma18
If every r satisfies
1
0
1
0|)(||)(| drrX
p
LPdrrY
ppxs
s LPyt
t
Y(r) = { t | yt≧ r }
y
r/p
x
X(r)
→ contradiction
max yttÎT
å
xssÎU
å ³ yt
xssÎS
å =1
|)(||)(| rXp
LPrY
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Summary19
max(deg(t) – θt +1)-approximation
Q. Better approximation?
P when θt = 1 for any t (maximum coverage)
when f is submodular, opt is a singleton
P when θt = deg(t) for any t (densest subhypergraph)
f is supermodular
Q. Fractional optimization for other problems?
Densest k-subgraph vs Densest subgraph
Q. Approx. for size-constrained problem?
LP for densest k-subhypergraph gives
deg(s)-approximation [Arulselvan14]max. f (X)
s.t. |X|≦k
maxf (X)
| X |
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Thank you for your attention