Submodular Maximization applied to Marketing Over Social...

Submodular Maximization applied toMarketing Over Social Networks

Vahab MirrokniGoogle Research, NYC

Marketing over Social NetworksI Online Social Networks: MySpace, Facebook.

I Monetizing Social Networks.I Viral Marketing: Word-of-Mouth Advertising.

I Users influence each others’ valuation on a social network.

I Marketing policy: In what order and at what price, dowe offer an item to buyers?

Marketing over Social Networks

I Online Social Networks: MySpace, Facebook.














Application 2: Guaranteed Banner AdvertisementI Online Advertisement: $20 billion annual revenue!I Banner Advertisement: 22% of the current revenue.

bjbi

Advertisers

Impressions

I Guaranteed Delivery: We pay a penalty for not meeting thedemand of each advertiser.

I Problem: Which set of advertisers should we commit to?


bjbi

Advertisers

Impressions




bjbi

Advertisers

Impressions




bjbi

Advertisers

Impressions



MarketingOver

Social Networks

GuaranteedBanner AdAllocation

RevenueMaximization

MarketingOver

Social Networks

GuaranteedBanner AdAllocation

RevenueMaximization

SubmodularMaximization

Outline

I Submodularity: Definitions, Applications.I Maximizing non-monotone Submodular Functions

I Approximation Algorithms.I Hardness Results.

I Application 1: Marketing over Social Networks

I Application 2: Guaranteed Banner ad Allocation.

Submodular FunctionsModel the law of diminishing returns and/or economies of scale.

Generalize Concave functions to set functions: f is definedon subsets of X .

DefinitionA set function f : 2X → R is submodular iff for any S ,T ,

f (S) + f (T ) ≥ f (S ∪ T ) + f (S ∩ T ).

Decreasing Marginal Value.

j

S

T

f is submodular, iff ∀S ⊂ T , j /∈ T :

f (S ∪ {j})− f (S) ≥ f (T ∪ {j})− f (T )

Submodular FunctionsModel the law of diminishing returns and/or economies of scale.Generalize Concave functions to set functions: f is definedon subsets of X .

S2 → f (S2)

S1 → f (S1)x

f (x) X

∆x∆x ∆x

∆f1

∆f2

∆f1 > ∆f2


f (S) + f (T ) ≥ f (S ∪ T ) + f (S ∩ T ).


j

S

T


f (S ∪ {j})− f (S) ≥ f (T ∪ {j})− f (T )



f (S) + f (T ) ≥ f (S ∪ T ) + f (S ∩ T ).


j

S

T


f (S ∪ {j})− f (S) ≥ f (T ∪ {j})− f (T )



f (S) + f (T ) ≥ f (S ∪ T ) + f (S ∩ T ).


j

S

T


f (S ∪ {j})− f (S) ≥ f (T ∪ {j})− f (T )

Examples of Submodular FunctionsI Simple Examples: Additive functions, Concave functions.

I Additive function: f (S) =∑

j∈S cj .I Concave function: f (S) = g(|S |) for a concave function g .

I Set coverage:f (S) = |⋃j∈S Aj |.

I Cuts in graphs and hypergraphs:f (S) = e(S ,S)

I Other places:I Maximum Facility location.I Utility functions in economics, Rank function of matroids.

Two categories:

I monotone functions, i.e, f (S) ≤ f (T ) for S ⊂ T .

I non-monotone functions: Cut functions, Maximum FacilityLocation.





A1

A3

A4

A5

A6

A2

S = {2, 3, 6}f (S) = |A2∪A3∪A6|



Two categories:I monotone functions, i.e, f (S) ≤ f (T ) for S ⊂ T .I non-monotone functions: Cut functions, Maximum Facility

Location.






S S

f (S) = e(S ,S)

S S

g(S) = e(S → S)


Two categories:I monotone functions, i.e, f (S) ≤ f (T ) for S ⊂ T .I non-monotone functions: Cut functions, Maximum Facility

Location.







Two categories:









Two categories:



Submodular MaximizationGiven: Value oracle, f : 2X → R is submodular.

S f (S)SubmodularOracle

Value Query

Goal: Maximize f (S) over all sets S ⊆ X , using small number ofqueries.

I Min-Cut is poly-time solvable.

I Submodular function minimization is poly-time solvable.[Schrijver, Fleischer-Fujishige-Iwata 2001].

I Max-Cut is NP-hard.

I α-approximation: output is at least α times OPT.

I For most of this talk, assume f : 2X → R+ is non-negative.



Value Query









Value Query









Value Query







Applications of Submodular Maximization

I Maximizing Monotone Submodular Functions (with cardinalityconstraint):

I Maximum Set Coverage.I Maximizing influence in social networks (Kempe, Kleinberg,

Tardos, KDD).I Optimal sensor installation for outbreak detection (LKGFVG,

KDD).

I Maximizing Non-monotone Submodular Functions.I Maximum Facility LocationI Segmentation Problems (Kleinberg, Papadimitriou, Raghavan,

JACM).I Least core value of general supermodular cost games (Schulz,

Uhan, APPROX).I Optimal marketing over social networks (Hartline, M.,

Sundararajan, WWW)I Revenue maximization for banner advertisement (Feige,

Immorlica, M., Nazerzadeh, WWW)

Applications of Submodular Maximization

I Maximizing Monotone Submodular Functions (with cardinalityconstraint):

I Maximum Set Coverage.I Maximizing influence in social networks (Kempe, Kleinberg,

Tardos, KDD).I Optimal sensor installation for outbreak detection (LKGFVG,

KDD).

I Maximizing Non-monotone Submodular Functions.I Maximum Facility LocationI Segmentation Problems (Kleinberg, Papadimitriou, Raghavan,

JACM).I Least core value of general supermodular cost games (Schulz,

Uhan, APPROX).I Optimal marketing over social networks (Hartline, M.,

Sundararajan, WWW)I Revenue maximization for banner advertisement (Feige,

Immorlica, M., Nazerzadeh, WWW)

Outline

I Submodularity.I Maximizing non-monotone Submodular Functions

I Approximation Algorithms.I Hardness Results.



Related work

I Maximizing monotone submodular functions with cardinalityconstraints, (|S | ≤ k): Greedy (1− 1/e)-approximation.[Nemhauser-Wolsey-Fisher ’78, Feige ’98]

I Maximizing non-monotone submodular functions has beenstudied in OR.[Lee-Nemhauser-Wang ’96, Goldengorkin-Tijssen-Tso’99]No guaranteed approximation factor known.

I For special cases, known approximation algorithms:I 0.878-approx for Max Cut [Goemans-Williamson ’95]I 0.874-approx for Max Di-Cut [Livnat-Lewin-Zwick ’02]I (1− 1/2k−1)-approx for Max Cut in k-uniform hypergraphs;I NP-hard to improve 7/8 approximation for k = 4 [Hastad ’01].

Related work




Related work




Our Results: Non-negative submodular functions

Feige, M., Vondrak (FOCS,07)

1. Approximation Algorithms for Maximizing non-monotonenon-negative submodular functions:Examples: Cut functions, Marketing over social Networks,core value of supermodular games.

I 0.33-approximation (deterministic local search).I 0.40-approximation (randomized ”smooth local search”).I 0.50-approximation for symmetric functions.

2. Hardness Results:

I For any fixed ε > 0, a (1/2 + ε)-approximation would requireexponentially many queries.

I Submodular functions with succinct representation: NP-hardto achieve (3/4 + ε)-approximation.




I 0.33-approximation (deterministic local search).

I 0.40-approximation (randomized ”smooth local search”).I 0.50-approximation for symmetric functions.







I 0.33-approximation (deterministic local search).I 0.40-approximation (randomized ”smooth local search”).

I 0.50-approximation for symmetric functions.

























Submodular Maximization: Local SearchLocal Operation:

I Add v : S ′ = S ∪ {v}.I Remove v : S ′ = S\{v}.

Improving local operation if f (S ′) > f (S).

1

2

3

S = {4}f (S) = 10

10 11 12

4

5

6

7

9

S




1

2

3

S = {4, 5}f (S) = 17

10 11 12

4

5

6

7

9

S




1

2

3

S = {4, 5, 7}f (S) = 23

10 11 12

4

5

6

7

9

S




1

2

3

S = {4, 5, 7, 3}f (S) = 27

10 11 12

4

5

6

7

9

S




1

2

3

S = {4, 7, 3}f (S) = 30

10 11 12

4

5

6

7

9

S




1

2

3

S = {4, 7, 3, 6}f (S) = 33

10 11 12

4

5

6

7

9

S




1

2

3

S = {4, 3, 6}f (L) = f (S) = 34

10 11 12

4

5

6

7

9

L = S

Output L or L.




Local Search Algorithm:

I Start with S = {v} where v is the singleton of max value.I While there is an improving local operation,

1. Perform a local operation s.t. f (S ′) > (1 + ε/n2)f (S).

I Return the better of f (S) and f (S).

Theorem (Feige, M., Vondrak)

The Local Search Algorithm returns at least (13 − ε

n )OPT ;for symmetric functions, at least (1

2 − εn )OPT (tight analysis).

A Structure Lemma

LemmaFor a local optimal solution L, and any subset C ⊂ L or a supersetL ⊂ C , f (L) ≥ f (C ).

If T0 ⊂ T1 ⊂ T2 ⊂ · · · ⊂ Tk = L ⊂ Tk+1 ⊂ · · · ⊂ Tn, then,

f (T0) ≤ f (T1) · · · ≤ f (Tk−1) ≤ f (L) ≥ f (Tk+1) ≥ . . . ≥ f (Tn)

Proof: Base of Induction: f (Tk−1) ≤ f (L) ≥ f (Tk+1).If Ti+1\Ti = ai , then by submodularity and local optimality of L:

f (Ti+1)− f (Ti ) ≥ f (L)− f (L\{ai}) ≥ 0⇒ f (Ti+1) ≥ f (Ti )

Therefore,f (T0) ≤ f (T1) ≤ f (T2) · · · ≤ f (L)

A Structure Lemma




T2 Tk = LT3T1 Tk−1

a2

ak

a1

a3

Tn = X




A Structure Lemma







Proof of The Structure Lemma



n )OPT .

Proof:Find a local optimum L and return either L or L. Consider theactual optimum C . By structure lemma,

I f (L) ≥ f (C ∩ L)

I f (L) ≥ f (C ∪ L)

Hence, again by submodularity,

2f (L) + f (L) ≥ f (C ∩ L) + f (C ∪ L) + f (L)

≥ f (C ∩ L) + f (C ∩ L) + f (X )

≥ f (C ) + f (∅) ≥ OPT .

Consequently, either f (L) or f (L) must be at least 13OPT .

Proof of The Structure Lemma



n )OPT .

Proof:Find a local optimum L and return either L or L. Consider theactual optimum C . By structure lemma,

I f (L) ≥ f (C ∩ L)

I f (L) ≥ f (C ∪ L)

Hence, again by submodularity,

2f (L) + f (L) ≥ f (C ∩ L) + f (C ∪ L) + f (L)

≥ f (C ∩ L) + f (C ∩ L) + f (X )

≥ f (C ) + f (∅) ≥ OPT .

Consequently, either f (L) or f (L) must be at least 13OPT .

Our Results

Feige, M., Vondrak [FMV]

1. Approximation Algorithms for Maximizing non-negativesubmodular functions:

I 0.33-approximation (deterministic local search).I 0.40-approximation (randomized ”smooth local search”).I 0.50-approximation for symmetric functions


I A (1/2 + ε)-approximation would require exponentially manyqueries.


Random Subsets

A

A(p)

A(p) is a random subset of A:

each element is picked with prob-

ability p

Sampling lemma

E[f (A(p))] ≥ pf (A) + (1− p)f (∅)

The Smooth Local Search Algorithm

I For any set A, let RA = A(2/3) ∪ A(1/3), a random set.

A

RARA is a random

set with some bias

in picking elements

from A

Let Φ(A) = E[f (RA)] - a smoothed variant of f (A).

Algorithm:

I Perform local search with respect to Φ(A).

I When a local optimum is found, return RA or A.


The Smooth Local Search algorithm returns at least 0.40OPT .



A

RARA is a random

set with some bias

in picking elements

from A

Let Φ(A) = E[f (RA)] - a smoothed variant of f (A).Algorithm:







A

RARA is a random

set with some bias

in picking elements

from A

Let Φ(A) = E[f (RA)] - a smoothed variant of f (A).Algorithm:





Proof Sketch of the Algorithm

Analysis: more complicated,using the sampling lemma for 3 sets. Let

I A = local optimum found by our algorithm

I R = A(2/3) ∪ A(1/3), our random set

I C = optimum

Main claims:

1. E[f (R)] ≥ E[f (R ∪ (A ∩ C ))].

2. E[f (R)] ≥ E[f (R ∩ (A ∪ C ))].

3. 920E[f (R∪(A∩C ))]+ 9

20E[f (R∩(A∪C ))]+ 110 f (A) ≥ 0.4OPT .

Our Results

1. Approximation Algorithms for Maximizing non-monotonesubmodular functions:



I It is impossible to improve the factor (1/2).I A (1/2 + ε)-approximation would require exponentially many

queries.I Submodular functions with succinct representation: NP-hard

to achieve (3/4 + ε)-approximation.

Proof Idea for the Hardness ResultGoal: Find two functions f and g which look identical to a typicalquery with high probability, and max g(S)

.= 2 max f (S).

Consider two functions f1, g1 : [0, 1]2 → R+:

1. f1(x , y) = (x + y)(2− x − y).2. g1(x , y) = 2x(1− y) + 2(1− x)y .

Observe: f1(x , x) = g1(x , x) ⇒ modify g1(x , y) to g2(x , y) s.t.f1(x , y) = g2(x , y) for |x − y | < ε.

Mapping to set functions: Let X = X1 ∪ X2, and

f (S) = f1(|S∩X1||X1| ,

|S∩X2||X2|

), and g(S) = g2

(|S∩X1||X1| ,

|S∩X2||X2|

).


.= 2 max f (S).


1. f1(x , y) = (x + y)(2− x − y).2. g1(x , y) = 2x(1− y) + 2(1− x)y .



f (S) = f1(|S∩X1||X1| ,

|S∩X2||X2|

), and g(S) = g2

(|S∩X1||X1| ,

|S∩X2||X2|

).


.= 2 max f (S).


1. f1(x , y) = (x + y)(2− x − y). (”complete graph cut”)2. g1(x , y) = 2x(1− y) + 2(1− x)y . (”bipartite graph cut”)



f (S) = f1(|S∩X1||X1| ,

|S∩X2||X2|

), and g(S) = g2

(|S∩X1||X1| ,

|S∩X2||X2|

).

S x y

X1 X2

1− x 1− y

Similar Technique for Combinatorial Auctions

I Using a similar technique, we can show information theoreticlower bounds for combinatorial auctions.

Buyers

f3(S3) f4(S4) f5(S5) f6(S6)f1(S1) f2(S2)

S1 S5 S6S4S3S2

I Goal: Partition items to maximize social welfare, i.e,∑

i fi (Si ).

Theorem (M., Schapira, Vondrak, EC08)

Achieving factor better than 1− 1e needs exponential number of

value queries.

Extra Constraints

I Cardinality Constraints: |S | ≤ k .

I Knapsack Constaints:∑

i∈S wi ≤ C .

I Matroid Constraints.

I Known results: Monotone Submodular functions:I Matroid or cardinality constraints: 1− 1

e (NWF78, Vondrak08)I k Knapsack constraints: 1− 1

e . (Sviridenko01, KST09)I k Matroid constraints: 1

k+1 (NWF78).

I (Lee, M., Natarajan, Sviridenko (STOC 2009))Maximizing non-monotone submodular functions:

I One matroid or cardinality constraints: 14 .

I k Knapsack constraints: 15 .

I k Matroid constraints: 1k+2+ 1

k

.

I k Partition matroid constraints: 1k+1+ 1

k−1

.

Extra Constraints



i∈S wi ≤ C .

I Matroid Constraints.I Known results: Monotone Submodular functions:

I Matroid or cardinality constraints: 1− 1e (NWF78, Vondrak08)

I k Knapsack constraints: 1− 1e . (Sviridenko01, KST09)

I k Matroid constraints: 1k+1 (NWF78).





k

.


k−1

.

Extra Constraints



i∈S wi ≤ C .

I Matroid Constraints.I Known results: Monotone Submodular functions:

I Matroid or cardinality constraints: 1− 1e (NWF78, Vondrak08)

I k Knapsack constraints: 1− 1e . (Sviridenko01, KST09)

I k Matroid constraints: 1k+1 (NWF78).





k

.


k−1

.

Extra Constraints: Algorithms

Lee, M., Nagarajan, Sviridenko, STOC’09

I Local Search Algorithms.

I Cardinality constraints: 14 .

I Local Search with add, remove, and swap operations.

I k Knapsack or budget constraints: 15 .

1. Solve a fractional variant on small elements.2. Round the fractional solution for small elements.3. Output the better of the solution for small elements and large

elements.


k

.

I Local Search: Delete operation and more complicatedexchange operations.








elements.


k

.









elements.


k

.


Approximating EverywhereI Can we learn a submodular function by polynomial number of

queries?I After polynomial number of queries, construct an oracle that

approximate f by f ′?

Sk−1

SubmodularOracle

Sk

S2f(S1), . . . , f(Sk)

S1S1

S3

· · ·Approximate

Oracle

S f ′(S)

S1

I Goal: Approximate f by f ′?

Theorem (Goemans, Iwata, Harvey, M. SODA09)

Achieving factor better than√

n needs exponential number ofvalue queries even for rank functions of matroids.


After a polynomial number of value queries to a monotontesubmodular function, we can approximate the function everywherewithin O(

√n log n).




Sk−1

SubmodularOracle

Sk

S2f(S1), . . . , f(Sk)

S1S1

S3

· · ·Approximate

Oracle

S f ′(S)

S1







√n log n).




Sk−1

SubmodularOracle

Sk

S2f(S1), . . . , f(Sk)

S1S1

S3

· · ·Approximate

Oracle

S f ′(S)

S1







√n log n).

Outline

I Submodularity.

I Maximizing non-monotone Submodular Functions



Marketing over Social NetworksI Online Social Networks: MySpace, Facebook.



















Application 1: Marketing over Social NetworksI Online Social Networks: MySpace, Facebook.I Viral Marketing: Word-of-Mouth Advertising.

I Users influence each others’ valuation on a social network.I Especially for Networked Goods, like Zune, Sprint, or Verizon.

I Model the influence among users by submodular (or concave)functions.

I fi (S) = g(|Ni ∩ S |).




I fi (S) = g(|Ni ∩ S |).




I fi (S) = g(|Ni ∩ S |).

100

A B

C

($100,$110,$115,$118)




I fi (S) = g(|Ni ∩ S |).

110

A B

C

($100,$110,$115,$118)




I fi (S) = g(|Ni ∩ S |).

115

A B

C

($100,$110,$115,$118)




I fi (S) = g(|Ni ∩ S |).

118

A B

C

($100,$110,$115,$118)

Marketing Model

I Given: A prior (probability distribution) Pi (S) for thevaluation function for user i given that a set S of usersalready adopted the item.

I Goal: Design a Marketing Policy to maximize the expectedrevenue.

I Marketing policy: Visit buyers one by one and offer them aprice.

I Ordering of buyers.I Pricing at each step.

I Optimal (myopic) Pricing: Optimal price to maximize revenueat each step (ignoring the future influence).

I Let fi (S) be the optimal revenue from buyer i using theoptimal (myopic) price given that a set S of buyers havebought the item.

I We call this function fi the influence function.

Marketing Model








Marketing Model








Marketing Model








Marketing Model: ExampleI Marketing policy: In what order and at what price do we

offer a digital good to buyers?

I Each buyer has a monotone submodular influence function.

(60,70)(100,105)

(100,110,115,118)

(50,60,65)

100?




(60,70)(100,105)

(100,110,115,118)

(50,60,65)

100 70?




(60,70)(100,105)

(100,110,115,118)

(50,60,65)

70

65?

100




(60,70)(100,105)

(100,110,115,118)

(50,60,65)

100 70

65

118?

Marketing Model

Marketing policy: In what order and at what price, do weoffer a digital good to buyers?

I Given: A prior (probability distribution) Pi (S) for thevaluation of user i given that a set S of users already adoptedthe item.


I The problem is NP-hard.

I Hartline, M., and Sundararajan [HMS] (WWW)

Marketing Model

Marketing policy: In what order and at what price, do weoffer a digital good to buyers?

I Given: A prior (probability distribution) Pi (S) for thevaluation of user i given that a set S of users already adoptedthe item.


I The problem is NP-hard.

I Hartline, M., and Sundararajan [HMS] (WWW)

Influence & Exploit Strategies

Influence & Exploit strategies:

1. Influence: Give the item for free to a set A of influentialbuyers.

2. Exploit: Apply the following strategy on the rest.I Optimal myopic pricing.I Random order.

Optimal (myopic) pricing:

I At each time, offer a price that maximizes the currentexpected revenue (ignoring the future influence).

Influence & Exploit Strategies

Influence & Exploit strategies:

1. Influence: Give the item for free to a set A of influentialbuyers.

2. Exploit: Apply the following strategy on the rest.I Optimal myopic pricing.I Random order.

Optimal (myopic) pricing:

I At each time, offer a price that maximizes the currentexpected revenue (ignoring the future influence).

Q1: How good are influence & exploit strategies?

Theorem (Hartline, M., Sundararajan)

Given monotone submodular influence functions, Influence &Exploit strategies give constant-factor approximations to theoptimal revenue.

Constant factor:

I 0.25 for the general distributions,

I 0.306 for distributions satisfying a monotone hazard-ratecondition,

I 0.33 for additive settings and uniform distribution,

I 0.66 for undirected graphs and uniform distributions,

I 0.94 for complete undirected graphs and uniform distributions.

Q1: How good are influence & exploit strategies?


Given monotone submodular influence functions, Influence &Exploit strategies give constant-factor approximations to theoptimal revenue.

Constant factor:

I 0.25 for the general distributions,

I 0.306 for distributions satisfying a monotone hazard-ratecondition,

I 0.33 for additive settings and uniform distribution,

I 0.66 for undirected graphs and uniform distributions,

I 0.94 for complete undirected graphs and uniform distributions.

Q2: How to find influential users?

I Question 2: How do we choose a set A of influential users togive the item for free to maximize the revenue?

I Let g(A) be the expected revenue, if the initial set is A.

I For some small sets A, g(∅) ≤ g(A).

I If we give the item for free to all users, g(X ) = 0.


Given monotone submodular influence functions, the revenuefunction g is a non-monotone non-negative submodular function.

I Thus, to find a set A that maximizes the revenue g(A), wecan use the 0.4-approximation local search algorithm fromFeige, M., Vondrak.

























Future Directions

I Marketing over Social Networks

I Avoiding Price Discremenation (Fixed-Price).I Iterative Pricing with Positive Network Externalities

(Akhlaghpour, Ghodsi, Haghpanah, Mahini, M., Nikzad).

I Cascading Effect and Influence Propagation.I Revenue Maximization for Fixed-Priced Marketing with

Influence Propagation (M., Sundararajna, Roch).

I Learning vs. Marketing.

Thank You

Outline

I Submodularity.

I Maximizing non-monotone Submodular Functions

I Application 1: Marketing over Social Networks.


Application 2: Guaranteed Banner AdvertisementOnline Advertisement: $20 billion annual revenue!Banner Advertisement: 22% of the current revenue.

Guaranteed Banner Advertisement:I Each advertiser i

I Interested in set Si of impressions, (e.g, young women inSeattle),

I Bids bi for each impression,I Needs di impressions,I Penalty αbi for not satisfying each unit (Guaranteed Delivery).

Sjdi = 4

bj

Si

bi

Advertisers

Impressions

(Si, bi, di)

Goal: Choose a set T of advertisers to maximize revenue, f (T ).If we give q items to advertiser i , we get

qbi − αbi (di − q) = q(1 + α)bi − diαbi

Application 2: Guaranteed Banner AdvertisementOnline Advertisement: $20 billion annual revenue!Banner Advertisement: 22% of the current revenue.Guaranteed Banner Advertisement:

I Each advertiser iI Interested in set Si of impressions, (e.g, young women in

Seattle),I Bids bi for each impression,I Needs di impressions,I Penalty αbi for not satisfying each unit (Guaranteed Delivery).

Sjdi = 4

bj

Si

bi

Advertisers

Impressions

(Si, bi, di)






Sjdi = 4

bj

Si

bi

Advertisers

Impressions

(Si, bi, di)

Goal: Choose a set T of advertisers to maximize revenue, f (T ).

If we give q items to advertiser i , we get





Sjdi = 4

bj

Si

bi

Advertisers

Impressions

(Si, bi, di)



Guaranteed Banner Advertisement: Submodularity


qbi − αbi (di − q) = q(1 + α)bi − diαbi = q(1 + α)bi − ci .

If we commit to set T of advertisers:

f (T ) = P(T )−∑i∈T ci = P(T )− C (T ).

Advertisers

Impressions

T

Maximum weighted matching for this graph.

(1 + α)bj(1 + α)bi

P (T ) =

P is submodular ⇒ f is submodular, but it can be negative.In fact, we prove that the problem is not approximable within anyconstant factor for any α = c .





f (T ) = P(T )−∑i∈T ci = P(T )− C (T ).

Advertisers

Impressions

T


(1 + α)bj(1 + α)bi

P (T ) =






f (T ) = P(T )−∑i∈T ci = P(T )− C (T ).

Advertisers

Impressions

T


(1 + α)bj(1 + α)bi

P (T ) =

P is submodular ⇒ f is submodular, but it can be negative.

In fact, we prove that the problem is not approximable within anyconstant factor for any α = c .





f (T ) = P(T )−∑i∈T ci = P(T )− C (T ).

Advertisers

Impressions

T


(1 + α)bj(1 + α)bi

P (T ) =


Structural Approximation

Feige, Immorlica, M., Nazerzadeh [FIMN]There are many natural submodular functions of the form:A submodular profit function - additive cost function =P(S)− C (S).Examples:

I Maximum Facility Location Problem.

I Segmentation Problems.

I Guaranteed Banner Ad Problem.

The above problems are possibly negative, and not approximablewithin any multiplicative approximatin factor.Structural Approximation

I Approximation factor is a function of the structure of thesolution.

We design greedy and linear programming-based tight structuralapproximation algorithms for the above problems.






The above problems are possibly negative, and not approximablewithin any multiplicative approximatin factor.




















Guaranteed Banner Ad ProblemGreedy Algorithm:

1. S = ∅.2. At each step, add an advertiser i ∈ X\S that maximizes

P(S∪{i})−P(S)−ci

P(S∪{i})−P(S) if P(S ∪ {i})− P(S)− ci > 0.

LP-based Algorithm:I Solve a Configurational LP and Round it.

max∑

i∈A,Q⊆SiX Q

i dibi ((1 + α)( |Q|di− ln |Q|di

)− α)

s.t.∑

i∈A,Q∈Si :j∈S X Qi ≤ 1 ∀j ∈ U∑

Q∈SiX Q

i ≤ 1 ∀i ∈ A

X Qi ≥ 0 ∀i ∈ A, ∀Q ∈ Si

Theorem[Feige, Immorlica, M., Nazerzadeh] The greedy algorithmand the LP-based algorithm achieve tight structural approximationfor (capacitated) maximum facility location, segmentationproblems, and guaranteed banner advertisement problem.




P(S∪{i})−P(S) if P(S ∪ {i})− P(S)− ci > 0.


max∑

i∈A,Q⊆SiX Q


)− α)

s.t.∑


Q∈SiX Q

i ≤ 1 ∀i ∈ A

X Qi ≥ 0 ∀i ∈ A, ∀Q ∈ Si





P(S∪{i})−P(S) if P(S ∪ {i})− P(S)− ci > 0.


max∑

i∈A,Q⊆SiX Q


)− α)

s.t.∑


Q∈SiX Q

i ≤ 1 ∀i ∈ A

X Qi ≥ 0 ∀i ∈ A, ∀Q ∈ Si


Future Directions

I Guaranteed Banner ad Allocation

I Uncertainty in Supply (impressions).I Uncertainty in Demand (advertisers).I Online Allocation Mechanisms.I Truthful Mechanisms.

Marketing overSocial Networks

Algorithmic & Economic aspects of the Internet

InternetMonetization(Stochastic

Optimization,Linear

Programming)

Search &Large Networks

AlgorithmicGame Theory

(Auctions,Mechanism

Design)

Guaranteed BannerAdvertisement




Optimization,Linear

Programming)


(Refined RandomWalks, LocalClustering)


(Auctions,Mechanism

Design)



Trust-based Recommendation Systems

Overlapping Clustering for Distributed Comp.



Optimization,Linear

Programming)


(Refined RandomWalks, LocalClustering)

PageRank Contributions & Link Spam Detection


(Auctions,Mechanism

Design)


Algorithms for Search and Large Networks

I Local Computation of PageRank Contributions & LinkSpam Detection

I Axiomatic Approach to Trust-Based RecommendationSystems

I Overlapping Clustering for Distributed Computation









Thank You

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