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Online Matching with Stochastic Rewards
"... The online matching problem has received significant attention in recent years because of its connections to allocation problems in Internet advertising, crowdsourcing, etc. In these realworld applications, the typical goal is not to maximize the number of allocations; rather it is to maximize th ..."
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The online matching problem has received significant attention in recent years because of its connections to allocation problems in Internet advertising, crowdsourcing, etc. In these realworld applications, the typical goal is not to maximize the number of allocations; rather it is to maximize the number of “successful ” allocations, where success of an allocation is governed by a stochastic process which follows the allocation. To address such applications, we propose and study the online matching problem with stochastic rewards (called the ONLINE STOCHASTIC MATCHING problem) in this paper. Our problem also has close connections to the existing literature on stochastic packing problems; in fact, our work initiates the study of online stochastic packing problems. We give a deterministic algorithm for the ONLINE STOCHASTIC MATCHING problem whose competitive ratio converges to (approximately) 0.567 for uniform and vanishing probabilities. We also give a randomized algorithm which outperforms the deterministic algorithm for higher probabilities. Finally, we complement our algorithms by giving an upper bound on the competitive ratio of any algorithm for this problem. This result shows that the best achievable competitive ratio for the ONLINE STOCHASTIC MATCHING problem is provably worse than that for the (nonstochastic) online matching problem.
Communication Complexity of Combinatorial Auctions with Submodular Valuations
"... We prove the first communication complexity lower bound for constantfactor approximation of the submodular welfare problem. More precisely, we show that a (1 − 1 2e +ɛ)approximation ( ≃ 0.816) for welfare maximization in combinatorial auctions with submodular valuations would require exponential c ..."
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We prove the first communication complexity lower bound for constantfactor approximation of the submodular welfare problem. More precisely, we show that a (1 − 1 2e +ɛ)approximation ( ≃ 0.816) for welfare maximization in combinatorial auctions with submodular valuations would require exponential communication. We also show NPhardness of (1 − 1 2e +ɛ)approximation in a computational model where each valuation is given explicitly by a table of constant size. Both results rule out better than (1 − 1 2e)approximations in every oracle model with a separate oracle for each player, such as the demand oracle model. Our main tool is a new construction of monotone submodular functions that we call multipeak submodular functions. Roughly speaking, given a family of sets F, we construct a monotone submodular function f with a high value f(S) for every set S ∈ F (a “peak”), and a low value on every set that does not intersect significantly any set in F. We also study two other related problems: maxmin allocation (for which we also get hardness of
Online submodular welfare maximization: Greedy is optimal
"... We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2competitive for welfare maximization with coverage valuations, unless NP = RP. Since the Greedy algorithm is known to be 1/2competitive for monotone submodular valuations, of which coverage is a sp ..."
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We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2competitive for welfare maximization with coverage valuations, unless NP = RP. Since the Greedy algorithm is known to be 1/2competitive for monotone submodular valuations, of which coverage is a special case, this proves that Greedy provides the optimal competitive ratio. On the other hand, we prove that Greedy in a stochastic setting with i.i.d. items and valuations satisfying diminishing returns is (1 − 1/e)competitive, which is optimal even for coverage valuations, unless NP = RP. For online budgetadditive allocation, we prove that no algorithm can be 0.612competitive with respect to a natural LP which has been used previously for this problem. 1
AdCell: Ad Allocation in Cellular Networks
"... Abstract. With more than four billion usage of cellular phones worldwide, mobile advertising has become an attractive alternative to online advertisements. In this paper, we propose a new targeted advertising policy for Wireless Service Providers (WSPs) via SMS or MMS namely AdCell. In our model, a ..."
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Abstract. With more than four billion usage of cellular phones worldwide, mobile advertising has become an attractive alternative to online advertisements. In this paper, we propose a new targeted advertising policy for Wireless Service Providers (WSPs) via SMS or MMS namely AdCell. In our model, a WSP charges the advertisers for showing their ads. Each advertiser has a valuation for specific types of customers in various times and locations and has a limit on the maximum available budget. Each query is in the form of time and location and is associated with one individual customer. In order to achieve a nonintrusive delivery, only a limited number of ads can be sent to each customer. Recently, new services have been introduced that offer locationbased advertising over cellular network that fit in our model (e.g., ShopAlerts by AT&T). We consider both online and offline version of the AdCell problem and develop approximation algorithms with constant competitive ratio. For the online version, we assume that the appearances of the queries follow a stochastic distribution and thus consider a Bayesian setting. Furthermore, queries may come from different
How to Sell Hyperedges: The Hypermatching Assignment Problem
, 2013
"... We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a khypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hyperm ..."
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We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a khypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hypermatching Assignment Problem (HAP) is to assign a set of pairwise disjoint bundles to clients so as to maximize the total profit while respecting the budgets. This problem has various applications in production planning and budgetconstrained auctions and generalizes wellstudied problems in combinatorial optimization: for example the weighted (unweighted) khypergraph matching problem is the special case of HAP with one client having unbounded budget and general (unit) profits; the Generalized Assignment Problem (GAP) is the special case of HAP with k = 1. Let ε> 0 denote an arbitrarily small constant. In this paper we obtain the following main results: • We give a randomized (k + 1 + ) approximation algorithm for HAP, which is based on rounding the 1round Lasserre strengthening of a novel LP. This is one of a few approximation results based on Lasserre hierarchies and our approach might be of independent interest. We remark that for weighted khypergraph matching no LP nor SDP relaxation is known to have integrality gap better than k − 1 + 1/k for general k [Chan and Lau, SODA’10]. • For the relevant special case that one wants to maximize the total revenue (i.e., bi,e = wi,e), we present a local search based (k + O( k))/2 approximation algorithm for k = O(1). This almost matches the best known (k + 1 + )/2 approximation ratio by Berman [SWAT’00] for
B.: Truthful mechanism design via correlated tree rounding
 In: Proceedings of the 16th ACM conference on Electronic commerce, ACM (2015
"... One of the most powerful algorithmic techniques for truthful mechanism design are maximalindistributionalrange (MIDR) mechanisms. Unfortunately, many algorithms using this paradigm rely on heavy algorithmic machinery and require the ellipsoid method or (approximate) solution of convex programs. ..."
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One of the most powerful algorithmic techniques for truthful mechanism design are maximalindistributionalrange (MIDR) mechanisms. Unfortunately, many algorithms using this paradigm rely on heavy algorithmic machinery and require the ellipsoid method or (approximate) solution of convex programs. In this paper, we present a simple and natural correlated rounding technique for designing mechanisms that are truthful in expectation. Our technique is elementary and can be implemented quickly. The main property we rely on is that the domain offers fractional optimum solutions with a tree structure. In auctions based on the generalized assignment problem, each bidder has a publicly known knapsack constraint that captures the subsets of items that are of value to him. He has a private valuation for each item and strives to maximize the value of assigned items minus payment. For this domain we design a mechanism for social welfare maximization. Our technique gives a truthful 2approximate MIDR mechanism without using the ellipsoid method or convex programming. In contrast to some previous work, our mechanism achieves exact truthfulness. In restrictedrelated scheduling with selfish machines, each job comes with a public weight, and it must be assigned to a machine from a public jobspecific subset. Each machine has a private speed and strives to maximize payments minus workload of jobs assigned to it. For this domain we design a mechanism
M.: A truthfulinexpectation mechanism for the generalized assignment problem. In: Web and Internet Economics
, 2014
"... mechanism for the generalized assignment auction. In such an auction, each bidder has a knapsack valuation function and bidders ’ values for items are private. We present a novel convex optimization program for the problem which makes a maximalindistributionalrange (MIDR) allocation rule. The pre ..."
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mechanism for the generalized assignment auction. In such an auction, each bidder has a knapsack valuation function and bidders ’ values for items are private. We present a novel convex optimization program for the problem which makes a maximalindistributionalrange (MIDR) allocation rule. The presented convex program contains at least (1 − 1 e) ratio of the optimal social welfare. We show how to implement the convex program in polynomial time using a fractional greedy algorithm which approximates the optimal solution within an arbitrarily small error. This leads to an approximately MIDR allocation rule which in turn transforms to an approximately truthfulinexpectation mechanism. From an algorithmic point of view, our contribution has importance, as well; it outperforms the existing optimization algorithms for the GAP in terms of runtime while the approximation ratio is comparable to the best given approximation.
Limitations of deterministic auction design for correlated bidders
"... Abstract. The seminal work of Myerson (Mathematics of OR 81) characterizes incentivecompatible singleitem auctions among bidders with independent valuations. In this setting, relatively simple deterministic auction mechanisms achieve revenue optimality. When bidders have correlated valuations, des ..."
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Abstract. The seminal work of Myerson (Mathematics of OR 81) characterizes incentivecompatible singleitem auctions among bidders with independent valuations. In this setting, relatively simple deterministic auction mechanisms achieve revenue optimality. When bidders have correlated valuations, designing the revenueoptimal deterministic auction is a computationally demanding problem; indeed, Papadimitriou and Pierrakos (STOC 11) proved that it is APXhard, obtaining an explicit inapproximability factor of 99.95%. In the current paper, we strengthen this inapproximability factor to 57/58 ≈ 98.3%. Our proof is based on a gappreserving reduction from the problem of maximizing the number of satisfied linear equations in an overdetermined system of linear equations modulo 2 and uses the classical inapproximability result of H˚astad (J. ACM 01). We furthermore show that the gap between the revenue of deterministic and randomized auctions can be as low as 13/14 ≈ 92.9%, improving an explicit gap of 947/948 ≈ 99.9 % by Dobzinski, Fu, and Kleinberg (STOC 11). 1
On the Complexity of Computing an Equilibrium in Combinatorial Auctions
, 2015
"... We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders ar ..."
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We study combinatorial auctions where each item is sold separately but simultaneously via a second price auction. We ask whether it is possible to efficiently compute in this game a pure Nash equilibrium with social welfare close to the optimal one. We show that when the valuations of the bidders are submodular, in many interesting settings (e.g., constant number of bidders, budget additive bidders) computing an equilibrium with good welfare is essentially as easy as computing, completely ignoring incentives issues, an allocation with good welfare. On the other hand, for subadditive valuations, we show that computing an equilibrium requires exponential communication. Finally, for XOS (a.k.a. fractionally subadditive) valuations, we show that if there exists an efficient algorithm that finds an equilibrium, it must use techniques that are very different from our current ones.