Our Results: Non-negative submodular functions Feige, M., Vondrak ( FOCS,07 ) 1. Approximation Algorithms for Maximizing non-monotone non-negative submodular functions: Examples: Cut functions, Marketing over social Networks, core value of supermodular games. ◮ 0 . 33-approximation (deterministic local search).
Our Results: Non-negative submodular functions Feige, M., Vondrak ( FOCS,07 ) 1. Approximation Algorithms for Maximizing non-monotone non-negative submodular functions: Examples: Cut functions, Marketing over social Networks, core value of supermodular games. ◮ 0 . 33-approximation (deterministic local search). ◮ 0 . 40-approximation (randomized ”smooth local search”).
Our Results: Non-negative submodular functions Feige, M., Vondrak ( FOCS,07 ) 1. Approximation Algorithms for Maximizing non-monotone non-negative submodular functions: Examples: Cut functions, Marketing over social Networks, core value of supermodular games. ◮ 0 . 33-approximation (deterministic local search). ◮ 0 . 40-approximation (randomized ”smooth local search”). ◮ 0 . 50-approximation for symmetric functions.
Our Results: Non-negative submodular functions Feige, M., Vondrak ( FOCS,07 ) 1. Approximation Algorithms for Maximizing non-monotone non-negative submodular functions: Examples: Cut functions, Marketing over social Networks, core value of supermodular games. ◮ 0 . 33-approximation (deterministic local search). ◮ 0 . 40-approximation (randomized ”smooth local search”). ◮ 0 . 50-approximation for symmetric functions. 2. Hardness Results: ◮ For any fixed ǫ > 0, a (1 / 2 + ǫ )-approximation would require exponentially many queries.
Our Results: Non-negative submodular functions Feige, M., Vondrak ( FOCS,07 ) 1. Approximation Algorithms for Maximizing non-monotone non-negative submodular functions: Examples: Cut functions, Marketing over social Networks, core value of supermodular games. ◮ 0 . 33-approximation (deterministic local search). ◮ 0 . 40-approximation (randomized ”smooth local search”). ◮ 0 . 50-approximation for symmetric functions. 2. Hardness Results: ◮ For any fixed ǫ > 0, a (1 / 2 + ǫ )-approximation would require exponentially many queries. ◮ Submodular functions with succinct representation: NP-hard to achieve (3 / 4 + ǫ )-approximation.
Submodular Maximization: Local Search Local Operation: ◮ Add v : S ′ = S ∪ { v } . ◮ Remove v : S ′ = S \{ v } . 10 11 12 1 4 6 S 3 9 2 5 7 S = { 4 } f ( S ) = 10
Submodular Maximization: Local Search Local Operation: ◮ Add v : S ′ = S ∪ { v } . ◮ Remove v : S ′ = S \{ v } . Improving local operation if f ( S ′ ) > f ( S ). 10 11 12 S 1 4 6 3 9 2 5 7 S = { 4 , 5 } f ( S ) = 17
Submodular Maximization: Local Search Local Operation: ◮ Add v : S ′ = S ∪ { v } . ◮ Remove v : S ′ = S \{ v } . Improving local operation if f ( S ′ ) > f ( S ). 10 11 12 S 1 4 6 3 9 2 5 7 S = { 4 , 5 , 7 } f ( S ) = 23
Submodular Maximization: Local Search Local Operation: ◮ Add v : S ′ = S ∪ { v } . ◮ Remove v : S ′ = S \{ v } . Improving local operation if f ( S ′ ) > f ( S ). 10 11 12 S 1 4 6 3 9 2 5 7 S = { 4 , 5 , 7 , 3 } f ( S ) = 27
Submodular Maximization: Local Search Local Operation: ◮ Add v : S ′ = S ∪ { v } . ◮ Remove v : S ′ = S \{ v } . Improving local operation if f ( S ′ ) > f ( S ). 10 11 12 S 1 4 6 3 9 2 5 7 S = { 4 , 7 , 3 } f ( S ) = 30
Submodular Maximization: Local Search Local Operation: ◮ Add v : S ′ = S ∪ { v } . ◮ Remove v : S ′ = S \{ v } . Improving local operation if f ( S ′ ) > f ( S ). 10 11 12 S 1 4 6 3 9 2 5 7 S = { 4 , 7 , 3 , 6 } f ( S ) = 33
Submodular Maximization: Local Search Local Operation: ◮ Add v : S ′ = S ∪ { v } . ◮ Remove v : S ′ = S \{ v } . Improving local operation if f ( S ′ ) > f ( S ). 10 11 12 L = S 1 4 6 3 9 2 5 7 Output L or ¯ L . S = { 4 , 3 , 6 } f ( L ) = f ( S ) = 34
Submodular Maximization: Local Search Local Operation: ◮ Add v : S ′ = S ∪ { v } . ◮ Remove v : S ′ = S \{ v } . Improving local operation if f ( S ′ ) > f ( S ). Local Search Algorithm: ◮ Start with S = { v } where v is the singleton of max value. ◮ While there is an improving local operation, 1. Perform a local operation s.t. f ( S ′ ) > (1 + ǫ/ n 2 ) f ( S ) . ◮ Return the better of f ( S ) and f ( S ). Theorem (Feige, M., Vondrak) The Local Search Algorithm returns at least ( 1 3 − ǫ n ) OPT; for symmetric functions, at least ( 1 2 − ǫ n ) OPT (tight analysis).
A Structure Lemma Lemma For a local optimal solution L, and any subset C ⊂ L or a superset L ⊂ C, f ( L ) ≥ f ( C ) .
A Structure Lemma Lemma For a local optimal solution L, and any subset C ⊂ L or a superset L ⊂ C, f ( L ) ≥ f ( C ) . If T 0 ⊂ T 1 ⊂ T 2 ⊂ · · · ⊂ T k = L ⊂ T k +1 ⊂ · · · ⊂ T n , then, f ( T 0 ) ≤ f ( T 1 ) · · · ≤ f ( T k − 1 ) ≤ f ( L ) ≥ f ( T k +1 ) ≥ . . . ≥ f ( T n ) a k a 3 a 2 a 1 T n = X T k − 1 T k = L T 1 T 2 T 3
A Structure Lemma Lemma For a local optimal solution L, and any subset C ⊂ L or a superset L ⊂ C, f ( L ) ≥ f ( C ) . If T 0 ⊂ T 1 ⊂ T 2 ⊂ · · · ⊂ T k = L ⊂ T k +1 ⊂ · · · ⊂ T n , then, f ( T 0 ) ≤ f ( T 1 ) · · · ≤ f ( T k − 1 ) ≤ f ( L ) ≥ f ( T k +1 ) ≥ . . . ≥ f ( T n ) Proof: Base of Induction: f ( T k − 1 ) ≤ f ( L ) ≥ f ( T k +1 ). If T i +1 \ T i = a i , then by submodularity and local optimality of L : f ( T i +1 ) − f ( T i ) ≥ f ( L ) − f ( L \{ a i } ) ≥ 0 ⇒ f ( T i +1 ) ≥ f ( T i ) Therefore, f ( T 0 ) ≤ f ( T 1 ) ≤ f ( T 2 ) · · · ≤ f ( L )
Proof of The Structure Lemma Theorem (Feige, M., Vondrak) The Local Search Algorithm returns at least ( 1 3 − ǫ n ) OPT. Proof: Find a local optimum L and return either L or L . Consider the actual optimum C . By structure lemma, ◮ f ( L ) ≥ f ( C ∩ L ) ◮ f ( L ) ≥ f ( C ∪ L )
Proof of The Structure Lemma Theorem (Feige, M., Vondrak) The Local Search Algorithm returns at least ( 1 3 − ǫ n ) OPT. Proof: Find a local optimum L and return either L or L . Consider the actual optimum C . By structure lemma, ◮ f ( L ) ≥ f ( C ∩ L ) ◮ f ( L ) ≥ f ( C ∪ L ) Hence, again by submodularity, 2 f ( 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 1 3 OPT .
Our Results Feige, M., Vondrak [FMV] 1. Approximation Algorithms for Maximizing non-negative submodular functions: ◮ 0 . 33-approximation (deterministic local search). ◮ 0 . 40-approximation (randomized ”smooth local search”). ◮ 0 . 50-approximation for symmetric functions 2. Hardness Results: ◮ A (1 / 2 + ǫ )-approximation would require exponentially many queries. ◮ Submodular functions with succinct representation: NP-hard to achieve (3 / 4 + ǫ )-approximation.
Random Subsets A A ( p ) is a random subset of A : each element is picked with prob- ability p A ( p ) Sampling lemma E [ f ( A ( p ))] ≥ pf ( A ) + (1 − p ) f ( ∅ )
The Smooth Local Search Algorithm ◮ For any set A , let R A = A (2 / 3) ∪ A (1 / 3), a random set. R A R A is a random set with some bias A in picking elements from A Let Φ( A ) = E [ f ( R A )] - a smoothed variant of f ( A ).
The Smooth Local Search Algorithm ◮ For any set A , let R A = A (2 / 3) ∪ A (1 / 3), a random set. R A R A is a random set with some bias A in picking elements from A Let Φ( A ) = E [ f ( R A )] - a smoothed variant of f ( A ). Algorithm: ◮ Perform local search with respect to Φ( A ). ◮ When a local optimum is found, return R A or A .
The Smooth Local Search Algorithm ◮ For any set A , let R A = A (2 / 3) ∪ A (1 / 3), a random set. R A R A is a random set with some bias A in picking elements from A Let Φ( A ) = E [ f ( R A )] - a smoothed variant of f ( A ). Algorithm: ◮ Perform local search with respect to Φ( A ). ◮ When a local optimum is found, return R A or A . Theorem (Feige, M., Vondrak) The Smooth Local Search algorithm returns at least 0 . 40 OPT.
Proof Sketch of the Algorithm Analysis: more complicated , using the sampling lemma for 3 sets. Let ◮ A = local optimum found by our algorithm ◮ R = A (2 / 3) ∪ A (1 / 3), our random set ◮ C = optimum Main claims: 1. E [ f ( R )] ≥ E [ f ( R ∪ ( A ∩ C ))]. 2. E [ f ( R )] ≥ E [ f ( R ∩ ( A ∪ C ))]. 20 E [ f ( R ∪ ( A ∩ C ))]+ 9 9 20 E [ f ( R ∩ ( A ∪ C ))]+ 1 3. 10 f ( A ) ≥ 0 . 4 OPT .
Our Results 1. Approximation Algorithms for Maximizing non-monotone submodular functions: ◮ 0 . 33-approximation (deterministic local search). ◮ 0 . 40-approximation (randomized ”smooth local search”). ◮ 0 . 5-approximation for symmetric functions. 2. Hardness Results: ◮ It is impossible to improve the factor (1 / 2). ◮ A (1 / 2 + ǫ )-approximation would require exponentially many queries. ◮ Submodular functions with succinct representation: NP-hard to achieve (3 / 4 + ǫ )-approximation.
Proof Idea for the Hardness Result Goal: Find two functions f and g which look identical to a typical query with high probability, and max g ( S ) . = 2 max f ( S ).
Proof Idea for the Hardness Result Goal: Find two functions f and g which look identical to a typical query with high probability, and max g ( S ) . = 2 max f ( S ). Consider two functions f 1 , g 1 : [0 , 1] 2 → R + : 1. f 1 ( x , y ) = ( x + y )(2 − x − y ). 2. g 1 ( x , y ) = 2 x (1 − y ) + 2(1 − x ) y . Observe: f 1 ( x , x ) = g 1 ( x , x ) ⇒ modify g 1 ( x , y ) to g 2 ( x , y ) s.t. f 1 ( x , y ) = g 2 ( x , y ) for | x − y | < ǫ .
Proof Idea for the Hardness Result Goal: Find two functions f and g which look identical to a typical query with high probability, and max g ( S ) . = 2 max f ( S ). Consider two functions f 1 , g 1 : [0 , 1] 2 → R + : 1. f 1 ( x , y ) = ( x + y )(2 − x − y ). (”complete graph cut”) 2. g 1 ( x , y ) = 2 x (1 − y ) + 2(1 − x ) y . (”bipartite graph cut”) Observe: f 1 ( x , x ) = g 1 ( x , x ) ⇒ modify g 1 ( x , y ) to g 2 ( x , y ) s.t. f 1 ( x , y ) = g 2 ( x , y ) for | x − y | < ǫ . Mapping to set functions: Let X = X 1 ∪ X 2 , and � � � � | S ∩ X 1 | | S ∩ X 2 | | S ∩ X 1 | | S ∩ X 2 | f ( S ) = f 1 | X 1 | , , and g ( S ) = g 2 | X 1 | , . | X 2 | | X 2 | x y S 1 − x 1 − y X 2 X 1
Similar Technique for Combinatorial Auctions ◮ Using a similar technique, we can show information theoretic lower bounds for combinatorial auctions. Buyers S 5 S 6 S 1 S 2 S 3 S 4 f 1 ( S 1 ) f 2 ( S 2 ) f 4 ( S 4 ) f 6 ( S 6 ) f 3 ( S 3 ) f 5 ( S 5 ) ◮ Goal: Partition items to maximize social welfare, i.e, � i f i ( S i ). Theorem (M., Schapira, Vondrak, EC08 ) Achieving factor better than 1 − 1 e needs exponential number of value queries.
Extra Constraints ◮ Cardinality Constraints: | S | ≤ k . ◮ Knapsack Constaints: � i ∈ S w i ≤ C . ◮ Matroid Constraints.
Extra Constraints ◮ Cardinality Constraints: | S | ≤ k . ◮ Knapsack Constaints: � i ∈ S w i ≤ C . ◮ Matroid Constraints. ◮ Known results: Monotone Submodular functions: ◮ Matroid or cardinality constraints: 1 − 1 e (NWF78, Vondrak08) ◮ k Knapsack constraints: 1 − 1 e . (Sviridenko01, KST09) ◮ k Matroid constraints: 1 k +1 (NWF78).
Extra Constraints ◮ Cardinality Constraints: | S | ≤ k . ◮ Knapsack Constaints: � i ∈ S w i ≤ C . ◮ Matroid Constraints. ◮ Known results: Monotone Submodular functions: ◮ Matroid or cardinality constraints: 1 − 1 e (NWF78, Vondrak08) ◮ k Knapsack constraints: 1 − 1 e . (Sviridenko01, KST09) ◮ k Matroid constraints: 1 k +1 (NWF78). ◮ (Lee, M., Natarajan, Sviridenko (STOC 2009)) Maximizing non-monotone submodular functions: ◮ One matroid or cardinality constraints: 1 4 . ◮ k Knapsack constraints: 1 5 . ◮ k Matroid constraints: 1 k . k +2+ 1 ◮ k Partition matroid constraints: 1 k − 1 . 1 k +1+
Extra Constraints: Algorithms Lee, M., Nagarajan, Sviridenko, STOC’09 ◮ Local Search Algorithms. 1 ◮ Cardinality constraints: 4 . ◮ Local Search with add, remove, and swap operations.
Extra Constraints: Algorithms Lee, M., Nagarajan, Sviridenko, STOC’09 ◮ Local Search Algorithms. 1 ◮ Cardinality constraints: 4 . ◮ Local Search with add, remove, and swap operations. 1 ◮ k Knapsack or budget constraints: 5 . 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.
Extra Constraints: Algorithms Lee, M., Nagarajan, Sviridenko, STOC’09 ◮ Local Search Algorithms. 1 ◮ Cardinality constraints: 4 . ◮ Local Search with add, remove, and swap operations. 1 ◮ k Knapsack or budget constraints: 5 . 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. 1 ◮ k Matroid constraints: k . k +2+ 1 ◮ Local Search: Delete operation and more complicated exchange operations.
Approximating Everywhere ◮ Can we learn a submodular function by polynomial number of queries? ◮ After polynomial number of queries, construct an oracle that approximate f by f ′ ? S 1 S 1 S 1 Submodular f ( S 1 ) , . . . , f ( S k ) S 2 Oracle S 3 Approximate · · · Oracle S k − 1 f ′ ( S ) S S k ◮ Goal: Approximate f by f ′ ?
Approximating Everywhere ◮ Can we learn a submodular function by polynomial number of queries? ◮ After polynomial number of queries, construct an oracle that approximate f by f ′ ? S 1 S 1 S 1 Submodular f ( S 1 ) , . . . , f ( S k ) S 2 Oracle S 3 Approximate · · · Oracle S k − 1 f ′ ( S ) S S k ◮ Goal: Approximate f by f ′ ? Theorem (Goemans, Iwata, Harvey, M. SODA09 ) Achieving factor better than √ n needs exponential number of value queries even for rank functions of matroids.
Approximating Everywhere ◮ Can we learn a submodular function by polynomial number of queries? ◮ After polynomial number of queries, construct an oracle that approximate f by f ′ ? S 1 S 1 S 1 Submodular f ( S 1 ) , . . . , f ( S k ) S 2 Oracle S 3 Approximate · · · Oracle S k − 1 f ′ ( S ) S S k ◮ Goal: Approximate f by f ′ ? Theorem (Goemans, Iwata, Harvey, M. SODA09 ) Achieving factor better than √ n needs exponential number of value queries even for rank functions of matroids. Theorem (Goemans, Iwata, Harvey, M. SODA09 ) After a polynomial number of value queries to a monotonte submodular function, we can approximate the function everywhere within O ( √ n log n ) .
Outline ◮ Submodularity. ◮ Maximizing non-monotone Submodular Functions ◮ Application 1: Marketing over Social Networks ◮ Application 2: Guaranteed Banner ad Allocation.
Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook.
Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Monetizing Social Networks. ◮ Viral Marketing: Word-of-Mouth Advertising.
Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Monetizing Social Networks. ◮ Viral Marketing: Word-of-Mouth Advertising. ◮ Users influence each others’ valuation on a social network.
Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Monetizing Social Networks. ◮ Viral Marketing: Word-of-Mouth Advertising. ◮ Users influence each others’ valuation on a social network. ◮ Marketing policy: In what order and at what price, do we offer an item to buyers?
Application 1: Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Viral Marketing: Word-of-Mouth Advertising.
Application 1: Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Viral Marketing: Word-of-Mouth Advertising. ◮ Users influence each others’ valuation on a social network. ◮ Especially for Networked Goods, like Zune, Sprint, or Verizon.
Application 1: Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Viral Marketing: Word-of-Mouth Advertising. ◮ Users influence each others’ valuation on a social network. ◮ Especially for Networked Goods, like Zune, Sprint, or Verizon. ◮ Model the influence among users by submodular (or concave) functions. C B A 100 ( $100 ,$110,$115,$118)
Application 1: Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Viral Marketing: Word-of-Mouth Advertising. ◮ Users influence each others’ valuation on a social network. ◮ Especially for Networked Goods, like Zune, Sprint, or Verizon. ◮ Model the influence among users by submodular (or concave) functions. C B A 110 ($100, $110 ,$115,$118)
Application 1: Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Viral Marketing: Word-of-Mouth Advertising. ◮ Users influence each others’ valuation on a social network. ◮ Especially for Networked Goods, like Zune, Sprint, or Verizon. ◮ Model the influence among users by submodular (or concave) functions. ◮ f i ( S ) = g ( | N i ∩ S | ). C B A 115 ($100,$110, $115 ,$118)
Application 1: Marketing over Social Networks ◮ Online Social Networks: MySpace, Facebook. ◮ Viral Marketing: Word-of-Mouth Advertising. ◮ Users influence each others’ valuation on a social network. ◮ Especially for Networked Goods, like Zune, Sprint, or Verizon. ◮ Model the influence among users by submodular (or concave) functions. ◮ f i ( S ) = g ( | N i ∩ S | ). C B A 118 ($100,$110,$115, $118 )
Marketing Model ◮ Given: A prior (probability distribution) P i ( S ) for the valuation function for user i given that a set S of users already adopted the item. ◮ Goal: Design a Marketing Policy to maximize the expected revenue.
Marketing Model ◮ Given: A prior (probability distribution) P i ( S ) for the valuation function for user i given that a set S of users already adopted the item. ◮ Goal: Design a Marketing Policy to maximize the expected revenue. ◮ Marketing policy : Visit buyers one by one and offer them a price. ◮ Ordering of buyers. ◮ Pricing at each step.
Marketing Model ◮ Given: A prior (probability distribution) P i ( S ) for the valuation function for user i given that a set S of users already adopted the item. ◮ Goal: Design a Marketing Policy to maximize the expected revenue. ◮ Marketing policy : Visit buyers one by one and offer them a price. ◮ Ordering of buyers. ◮ Pricing at each step. ◮ Optimal (myopic) Pricing: Optimal price to maximize revenue at each step (ignoring the future influence).
Marketing Model ◮ Given: A prior (probability distribution) P i ( S ) for the valuation function for user i given that a set S of users already adopted the item. ◮ Goal: Design a Marketing Policy to maximize the expected revenue. ◮ Marketing policy : Visit buyers one by one and offer them a price. ◮ Ordering of buyers. ◮ Pricing at each step. ◮ Optimal (myopic) Pricing: Optimal price to maximize revenue at each step (ignoring the future influence). ◮ Let f i ( S ) be the optimal revenue from buyer i using the optimal (myopic) price given that a set S of buyers have bought the item. ◮ We call this function f i the influence function.
Marketing Model: Example ◮ Marketing policy: In what order and at what price do we offer a digital good to buyers? ◮ Each buyer has a monotone submodular influence function. (50,60,65) 100? ( 100 ,105) (60,70) (100,110,115,118)
Marketing Model: Example ◮ Marketing policy: In what order and at what price do we offer a digital good to buyers? ◮ Each buyer has a monotone submodular influence function. (50,60,65) 70? 100 (100,105) (60, 70 ) (100,110,115,118)
Marketing Model: Example ◮ Marketing policy: In what order and at what price do we offer a digital good to buyers? ◮ Each buyer has a monotone submodular influence function. (50,60, 65 ) 65? 70 100 (100,105) (60,70) (100,110,115,118)
Marketing Model: Example ◮ Marketing policy: In what order and at what price do we offer a digital good to buyers? ◮ Each buyer has a monotone submodular influence function. (50,60,65) 65 70 100 (100,105) (60,70) 118? (100,110,115, 118 )
Marketing Model Marketing policy: In what order and at what price, do we offer a digital good to buyers? ◮ Given: A prior (probability distribution) P i ( S ) for the valuation of user i given that a set S of users already adopted the item. ◮ Goal: Design a Marketing Policy to maximize the expected revenue.
Marketing Model Marketing policy: In what order and at what price, do we offer a digital good to buyers? ◮ Given: A prior (probability distribution) P i ( S ) for the valuation of user i given that a set S of users already adopted the item. ◮ Goal: Design a Marketing Policy to maximize the expected revenue. ◮ The problem is NP-hard. ◮ Hartline, M., and Sundararajan [HMS] ( WWW )
Influence & Exploit Strategies Influence & Exploit strategies : 1. Influence: Give the item for free to a set A of influential buyers . 2. Exploit: Apply the following strategy on the rest. ◮ Optimal myopic pricing. ◮ Random order.
Influence & Exploit Strategies Influence & Exploit strategies : 1. Influence: Give the item for free to a set A of influential buyers . 2. Exploit: Apply the following strategy on the rest. ◮ Optimal myopic pricing. ◮ Random order. Optimal (myopic) pricing: ◮ At each time, offer a price that maximizes the current expected 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 the optimal revenue.
Q1: How good are influence & exploit strategies? Theorem (Hartline, M., Sundararajan) Given monotone submodular influence functions, Influence & Exploit strategies give constant-factor approximations to the optimal revenue. Constant factor: ◮ 0 . 25 for the general distributions, ◮ 0 . 306 for distributions satisfying a monotone hazard-rate condition, ◮ 0 . 33 for additive settings and uniform distribution, ◮ 0 . 66 for undirected graphs and uniform distributions, ◮ 0 . 94 for complete undirected graphs and uniform distributions.
Q2: How to find influential users? ◮ Question 2: How do we choose a set A of influential users to give the item for free to maximize the revenue? ◮ Let g(A) be the expected revenue, if the initial set is A .
Q2: How to find influential users? ◮ Question 2: How do we choose a set A of influential users to give the item for free to maximize the revenue? ◮ Let g(A) be the expected revenue, if the initial set is A . ◮ For some small sets A , g ( ∅ ) ≤ g ( A ). ◮ If we give the item for free to all users, g ( X ) = 0.
Q2: How to find influential users? ◮ Question 2: How do we choose a set A of influential users to give the item for free to maximize the revenue? ◮ Let g(A) be the expected revenue, if the initial set is A . ◮ For some small sets A , g ( ∅ ) ≤ g ( A ). ◮ If we give the item for free to all users, g ( X ) = 0. Theorem (Hartline, M., Sundararajan) Given monotone submodular influence functions, the revenue function g is a non-monotone non-negative submodular function.
Q2: How to find influential users? ◮ Question 2: How do we choose a set A of influential users to give the item for free to maximize the revenue? ◮ Let g(A) be the expected revenue, if the initial set is A . ◮ For some small sets A , g ( ∅ ) ≤ g ( A ). ◮ If we give the item for free to all users, g ( X ) = 0. Theorem (Hartline, M., Sundararajan) Given monotone submodular influence functions, the revenue function g is a non-monotone non-negative submodular function. ◮ Thus, to find a set A that maximizes the revenue g ( A ), we can use the 0 . 4-approximation local search algorithm from Feige, M., Vondrak.
Future Directions ◮ Marketing over Social Networks ◮ Avoiding Price Discremenation (Fixed-Price). ◮ Iterative Pricing with Positive Network Externalities (Akhlaghpour, Ghodsi, Haghpanah, Mahini, M., Nikzad). ◮ Cascading Effect and Influence Propagation. ◮ Revenue Maximization for Fixed-Priced Marketing with Influence Propagation (M., Sundararajna, Roch). ◮ Learning vs. Marketing.
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