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
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
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
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
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
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
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) 1 2 3 1 2 3 1 2 3 l l l Uniform Price Strategies Slide 4
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
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
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
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
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
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
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
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
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
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
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
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
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
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 3. 3. 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
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 1 2 3 Linear externalities Maximization is NP-hard O(log m)-approx. algorithm ext i,j (l i,j ) 1 2 3 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 1 2 3 Uniform Price Strategies Slide 19
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