Welfare Maximization with Friends-of-Friends Network Externalities

Transcription

1 Welfare Maximization with Friends-of-Friends Network Externalities Extended version of a talk at STACS 2015, Munich Wolfgang Dvořák 1 joint work with: Sayan Bhattacharya 2, Monika Henzinger 1, Martin Starnberger 1 1 Theory and Applications of Algorithms Group, University of Vienna. 2 Institute of Mathematical Sciences, Chennai. March 7, 2015 Uniform Price Strategies Slide 1

2 1. Motivation & Problem Statement Network Externalities in Social Networks Network Externalities: In a network an agent s valuation for an item may depend on whether connected agents have the same item. Example: software with network functionality In previous works only direct neighbors (friends) are considered. Friends-of-friends Externalities: Typically the number of friends-of-friends is magnitudes larger than the number of friends. If a friend has the same item and a friend of him also has the item one might obtain externality from the friends-of-friend. In many online social networks friend-of-friends are still visible Beside friends we also consider friends-of-friends as source of externalities if there is a common friend with the same item. Uniform Price Strategies Slide 2

3 1. Motivation & Problem Statement Problem Statement Welfare Maximization with Friends-of-Friends Externalities Social Network (V,E): agents (vertices) have friendship-like relationships (edges) m different digital goods (e.g., and ) Unit demand: every agent can receive only one item Agents j s valuation for good i is ext i,j ( S i,j ) where S i,j consists of (a) j s friends who receive good i and (b) j s friends-of-friends who receive good i and share a friend with j who receives good i a b c d S,a = {} = 0 Goal: Find an assignment A of items to agents such that the social welfare j V ext A(j),j( S A(j),j ) is maximized. e Uniform Price Strategies Slide 3

4 1. Motivation & Problem Statement Problem Statement Welfare Maximization with Friends-of-Friends Externalities Social Network (V,E): agents (vertices) have friendship-like relationships (edges) m different digital goods (e.g., and ) Unit demand: every agent can receive only one item Agents j s valuation for good i is ext i,j ( S i,j ) where S i,j consists of (a) j s friends who receive good i and (b) j s friends-of-friends who receive good i and share a friend with j who receives good i a b c d S,a = {b} = 1 Goal: Find an assignment A of items to agents such that the social welfare j V ext A(j),j( S A(j),j ) is maximized. e Uniform Price Strategies Slide 3

5 1. Motivation & Problem Statement Problem Statement Welfare Maximization with Friends-of-Friends Externalities Social Network (V,E): agents (vertices) have friendship-like relationships (edges) m different digital goods (e.g., and ) Unit demand: every agent can receive only one item Agents j s valuation for good i is ext i,j ( S i,j ) where S i,j consists of (a) j s friends who receive good i and (b) j s friends-of-friends who receive good i and share a friend with j who receives good i a b c d S,a = {b} = 1 Goal: Find an assignment A of items to agents such that the social welfare j V ext A(j),j( S A(j),j ) is maximized. e Uniform Price Strategies Slide 3

6 1. Motivation & Problem Statement Problem Statement Welfare Maximization with Friends-of-Friends Externalities Social Network (V,E): agents (vertices) have friendship-like relationships (edges) m different digital goods (e.g., and ) Unit demand: every agent can receive only one item Agents j s valuation for good i is ext i,j ( S i,j ) where S i,j consists of (a) j s friends who receive good i and (b) j s friends-of-friends who receive good i and share a friend with j who receives good i a b c S,a = {b, c, e} = 3 Goal: Find an assignment A of items to agents such that the social welfare j V ext A(j),j( S A(j),j ) is maximized. d e Uniform Price Strategies Slide 3

7 1. Motivation & Problem Statement Problem Statement (cont.) Different kind of valuation functions concave externality ext i,j ( ) concave linear externality ext i,j ( ) linear s-step function externality ext i,j (l) is constant for l s and 0 otherwise ext i,j (l) ext i,j (l) ext i,j (l) l l l Uniform Price Strategies Slide 4

8 1. Motivation & Problem Statement Related work Bhalgat et al. (EC 2012) study Welfare Maximization with friends externalities: concave externalities: O(log n(log n + log m)) approximation for general graphs; step-function: (1 1/e)/16-approximation for 1-step functions linear externalities: O(1) approximation submodular externalities: O(log 3 n) approximation in the demand oracle model. convex externalities: 2 O(d) approximation if the function is bounded by a degree d polynomial. Uniform Price Strategies Slide 5

9 2. Concave externalities Concave externalities Results Theorem (APX-hardness) It is NP-hard to approximate social welfare under concave externalities better than (1 1 e ). By a reduction from MAX COVERAGE. Theorem The social welfare under concave externalities can be O( n) approximated. Uniform Price Strategies Slide 6

10 2. Concave externalities Algorithm for Concave externalities Idea: Either most of the social welfare comes 1. from agents j with S i,j n or 2. from agents j with S i,j < n. Case 1: a O( n) approximation is given by the optimal among the assignments that give the same item to all of the agents. (for (1) there are at most n different items) Case 2: a O( n) approximation is computed as follows. 1. Reduce the externalities to a 1-step functions externalities by setting êxt i,j ( S i,j ) to ext i,j (1) if S i,j 1 and 0 otherwise. 2. Then use a constant factor approximation for 1-step functions (e.g. Bhalgat et al. (EC 2012)) to solve the new instance. Our algorithm computes both assignments a picks the better one. Uniform Price Strategies Slide 7

11 3. Step function externalities Step function externalities - Results Theorem Maximizing social welfare under arbitrary s-step externalities is not approximable within O(n 1/2 ε ) unless NP=ZPP. By a reduction from Max Independent Set Theorem (APX-hardness) For fixed s 1, it is NP-hard to approximate social welfare under s-step function externalities better than (1 1 e ). By a reduction from MAX COVERAGE. Theorem The Social welfare under 1-step function externalities can be 1/2 (1 1/e) approximated. 2-step function externalities can be 1/6 (1 1/e) approximated. Uniform Price Strategies Slide 8

12 3. Step function externalities Algorithm for 2-step function externalities 1.Step: Compute a maximal collection T of mutually disjoint triples of connected nodes in the graph G. T 1 b B 1 f a d P 1 c e The graph G V \T, consists of a mutually disjoint collection of pairs P and a set of isolated nodes say B. Uniform Price Strategies Slide 9

13 3. Step function externalities Algorithm for 2-step function externalities An assignment A is consistent (with T ) if for each triple and each pair all agents in the tuple get the same item. T 1 b B 1 f T 1 b B 1 f a d P 1 a d P 1 c e c e Inconsistent assignment Consistent assignment Lemma The social welfare from the optimal consistent assignment is at least (1/6) Opt, where Opt is the optimum over all assignments. Proof by a randomized algorithm mapping the optimal assignment to a consistent assignment with expected welfare (1/6) Opt. Uniform Price Strategies Slide 10

14 3. Step function externalities Algorithm for 2-step function externalities 2.Step: Reduce the problem of computing consistent assignments to the following allocation problem. Consider the set of Triples T as clients The pairs and isolated nodes are resources R = P B, each of them must be assigned to exactly one client The utility u T (S T ) of T T for S T R is defined such that it corresponds to the optimal social welfare of the agents in T and S T when they all get the same item. a T 1 b c d P 1 B 1 f e Clients T = {T 1 } Resources T = {P 1, B 1 } u T1 ({P 1 })=max ext i,j (2) i I j {a,b,c,d,e} Uniform Price Strategies Slide 11

15 3. Step function externalities Algorithm for 2-step function externalities Theorem (Feige (2009)) If (1) the utility functions are fractionally subadditive and (2) there is a poly time demand oracle then the maximal social welfare of the allocation problem can be (1 1/e) approximated. Lemma The functions u T (.) are fractionally subadditive and there is a polynomial time demand oracle. 3.Step: Using Feige s algorithm to (1 1/e) approximate the above setting. Uniform Price Strategies Slide 12

16 4. Linear externalities Linear externalities Results Theorem (APX-hardness) It is NP-hard to compute the optimal social welfare under linear externalities. By a reduction from MAX COVERAGE Theorem The social welfare under concave externalities can be O(log m) approximated. Uniform Price Strategies Slide 13

17 4. Linear externalities Algorithm for Linear externalities Outline 1.Step: Build a random partition (V 1, V 2, V 3 ) of V and only consider edges between V 1 V 2 and V 2 V 3. New Goal: Maximize the social welfare of V 1 (Restricted-Welfare) 2.Step: Solve LP relaxation of the Restricted-Welfare Problem 3.Step: Preprocess the solution of the LP (losing O(log m)) 4.Step: Randomized Rounding Uniform Price Strategies Slide 14

18 4. Linear externalities Algorithm for Linear externalities 1.Step: Build a random partition (V 1, V 2, V 3 ) of V and only consider edges between V 1 V 2 and V 2 V 3. New Goal: Maximize the social welfare of V 1 (Restricted-Welfare) a b d V 1 a b V 2 V 3 d c e c e Theorem Any α-approx. for the Restricted-Welfare problem is also an O(α)-approx. for the welfare-maximization problem in general graphs with concave externalities. Uniform Price Strategies Slide 15

19 4. Linear externalities Algorithm for Linear externalities 2.Step: Solve LP relaxation of the Restricted-Welfare Problem Maximize: ( λ ij j V 1 i I k V 2 F 1 j α(i, j, k) + l V 3 F 2 j ) β(i, j, l) (1) β(i, j, l) min{w(i, l), y(i, j)} i I, j V 1, l V 3 F 2 j (2) β(i, j, l) k F 1 j F 1 l z(i, k) i I, j V 1, l V 3 F 2 j (3) α(i, j, k) min{y(i, j), z(i, k)} i I, j V 1, k V 2 Fj 1 (4) i y(i, j) 1, i z(i, k) 1, i w(i, l) 1 j, k, l (5) 0 y(i, j), z(i, k), w(i, l), α(i, j, k), β(i, j, l) i, j, k, l (6) y(i, j), z(i, k), w(i, l)... node j, k, l gets item i α(i, j, k)... k is friend of j and both get item i β(i, j, l)... l is friend-of-friend of j, both get item i and they have a common friend with item i Uniform Price Strategies Slide 16

20 4. Linear externalities Algorithm for Linear externalities 3.Step: Preprocess the solution of the LP Lemma In polynomial time, we can get a feasible solution that is an O(log m) approximation to the optimal objective, with α(i, j, k), β(i, j, l), y(i, j), w(i, l) {0, γ} for some γ [0, 1], z(i, k) γ Uniform Price Strategies Slide 17

21 4. Linear externalities Algorithm for Linear externalities 4.Step: Randomized Rounding 1. Start with the feasible solution to the LP given by the Lemma. 2. Set T 0, and W 0 V = V 1 V 2 V For all items i I = {1,..., m}: 3.1 Set W i W i 1 \ T i 1, and T i. 3.2 With probability γ/4 do For nodes with y(i, j) = γ set T i T i {j} For nodes with w(i, l) = γ set T i T i {l} For nodes k V 2 with probability z(i, k)/γ set T i T i {k} 3.3 Assign item i to all nodes in W i T i 4. Return assignment Theorem The rounding scheme gives an O(log m)-approximation to the Restricted-Welfare problem. Uniform Price Strategies Slide 18

22 5. Summary & Conclusion Summary of Results Concave externalities NP-hard to (1 1/e + ɛ)-approx. ext i,j (l i,j ) O( n)-approx. algorithm l i,j Linear externalities Maximization is NP-hard O(log m)-approx. algorithm ext i,j (l i,j ) l i,j 2-step function externalities NP-hard to (1 1/e + ɛ)-approx. ext i,j (l i,j ) 1/6 (1 1/e)-approx. algorithm l i,j Uniform Price Strategies Slide 19

23 5. Summary & Conclusion Future work Close the gaps between algorithms and hardness results Consider submodular externalities Friends-of-friends externalities in other scenarios Incentive-compatible social welfare maximizing auctions (with strategic agents) Revenue maximization when selling a product (in scenarios like Hartline et al. (WWW 08))... Thank you for your attention! Uniform Price Strategies Slide 20