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34
Maximizing a Monotone Submodular Function subject to a Matroid Constraint
, 2008
"... Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)app ..."
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Cited by 63 (0 self)
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Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f(S) for a given set S) [28], and also for explicitly posed instances assuming P � = NP [10]. In this paper, we provide a randomized (1 − 1/e)approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires X  to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.
Online stochastic packing applied to display ad allocation
 In ESA
, 2010
"... Inspired by online ad allocation, we study online stochastic packing linear programs from theoretical and practical standpoints. We first present a nearoptimal online algorithm for a general class of packing linear programs which model various online resource allocation problems including online va ..."
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Cited by 40 (4 self)
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Inspired by online ad allocation, we study online stochastic packing linear programs from theoretical and practical standpoints. We first present a nearoptimal online algorithm for a general class of packing linear programs which model various online resource allocation problems including online variants of routing, ad allocations, generalized assignment, and combinatorial auctions. As our main theoretical result, we prove that a simple primaldual trainingbased algorithm achieves a (1 − o(1))approximation guarantee in the random order stochastic model. This is a significant improvement over logarithmic or constantfactor approximations for the adversarial variants of the same problems (e.g. factor 1 − 1e for online ad allocation, and log(m) for online routing). We then focus on the online display ad allocation problem and study the efficiency and fairness of various trainingbased and online allocation algorithms on data sets collected from reallife display ad allocation system. Our experimental evaluation confirms the effectiveness of trainingbased primaldual algorithms on real data sets, and also indicate an intrinsic tradeoff between fairness and efficiency. 1
Limitations of randomized mechanisms for combinatorial auctions
 In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS
, 2011
"... Abstract — The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfaremaximization in combinatorial auctions. ..."
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Cited by 18 (4 self)
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Abstract — The design of computationally efficient and incentive compatible mechanisms that solve or approximate fundamental resource allocation problems is the main goal of algorithmic mechanism design. A central example in both theory and practice is welfaremaximization in combinatorial auctions. Recently, a randomized mechanism has been discovered for combinatorial auctions that is truthful in expectation and guarantees a (1 − 1/e)approximation to the optimal social welfare when players have coverage valuations [11]. This approximation ratio is the best possible even for nontruthful algorithms, assuming P ̸ = NP [16]. Given the recent sequence of negative results for combinatorial auctions under more restrictive notions of incentive compatibility [7], [2], [9], this development raises a natural question: Are truthfulinexpectation mechanisms compatible with polynomialtime approximation in a way that deterministic or universally truthful
The Submodular Welfare Problem with demand queries
, 2009
"... We consider the Submodular Welfare Problem where we have m items and n players with given utility functions wi: 2 [m] → R+. The utility functions are assumed to be monotone and submodular. We want to find an allocation of disjoint sets of items (S1, S2,..., Sn) maximizing i wi(Si). A (1 − 1/e)appr ..."
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Cited by 16 (3 self)
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We consider the Submodular Welfare Problem where we have m items and n players with given utility functions wi: 2 [m] → R+. The utility functions are assumed to be monotone and submodular. We want to find an allocation of disjoint sets of items (S1, S2,..., Sn) maximizing i wi(Si). A (1 − 1/e)approximation for this problem in the demand oracle model has been given by Dobzinski and Schapira [5]. We improve this algorithm by presenting a (1 − 1/e + ɛ)approximation for some small fixed ɛ> 0. We also show that the Submodular Welfare Problem is NPhard to approximate within a ratio better than some ρ < 1. Moreover, this holds even when for each player there are only a constant number of items that have nonzero utility. The constant size restriction on utility functions makes it easy for players to efficiently answer any ”reasonable ” query about their utility functions. In contrast, for classes of instances that were used for previous hardness of approximation results, we present an incentive compatible (in expectation) mechanism based on fair division queries that achieves an optimal solution. 1
Improved Approximation Algorithms for Budgeted Allocations
"... Abstract. We provide a 3/2approximation algorithm for an offline budgeted allocations problem, an improvement over the e/(e − 1) approximation of Andelman and Mansour [1] and the e/(e − 1) − ɛ approximation (for ɛ ≈ 0.0001) of Feige and Vondrak [5] for the more general Maximum Submodular Welfare ( ..."
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Cited by 13 (1 self)
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Abstract. We provide a 3/2approximation algorithm for an offline budgeted allocations problem, an improvement over the e/(e − 1) approximation of Andelman and Mansour [1] and the e/(e − 1) − ɛ approximation (for ɛ ≈ 0.0001) of Feige and Vondrak [5] for the more general Maximum Submodular Welfare (SMW) problem. For a special case of our problem, we improve this ratio to √ 2. Finally, we prove that it is APXhard. The problem we study has applications to sponsored search auctions. 1
Budgeted allocations in the fullinformation setting
 In: APPROX ’08 (LNCS 5171
, 2008
"... Abstract. We build on the work of Andelman & Mansour and Azar, Birnbaum, Karlin, Mathieu & Thach Nguyen to show that the fullinformation (i.e., offline) budgetedallocation problem can be approximated to within 4/3: we conduct a rounding of the natural LP relaxation, for which our algorithm ..."
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Cited by 12 (1 self)
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Abstract. We build on the work of Andelman & Mansour and Azar, Birnbaum, Karlin, Mathieu & Thach Nguyen to show that the fullinformation (i.e., offline) budgetedallocation problem can be approximated to within 4/3: we conduct a rounding of the natural LP relaxation, for which our algorithm matches the known lowerbound on the integrality gap. 1
A new approximation technique for resourceallocation problems
, 2009
"... Abstract: We develop a rounding method based on random walks in polytopes, which leads to improved approximation algorithms and integrality gaps for several assignment problems that arise in resource allocation and scheduling. In particular, it generalizes the work of Shmoys & Tardos on the gene ..."
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Cited by 9 (4 self)
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Abstract: We develop a rounding method based on random walks in polytopes, which leads to improved approximation algorithms and integrality gaps for several assignment problems that arise in resource allocation and scheduling. In particular, it generalizes the work of Shmoys & Tardos on the generalized assignment problem in two different directions, where the machines have hard capacities, and where some jobs can be dropped. We also outline possible applications and connections of this methodology to discrepancy theory and iterated rounding.
Approximability of sparse integer programs
 In Proc. 17th ESA
, 2009
"... The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ..."
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Cited by 9 (1 self)
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The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ǫ> 0, if P = NP this ratio cannot be improved to k − 1 − ǫ, and under the unique games conjecture this ratio cannot be improved to k − ǫ. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsackcover inequalities. Second, for packing integer programs {max cx: Ax ≤ b,0 ≤ x ≤ d} where A has at most k nonzeroes per column, we give a 2 k k 2approximation algorithm. This is the first polynomialtime approximation algorithm for this problem with approximation ratio depending only on k, for any k> 1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution. Note added after publication: This version includes subsequent developments: a O(k 2) approximation for the latter problem using the iterated rounding framework, and several literature reference updates including a O(k)approximation for the same problem by Bansal et al.
Selective call out and real time bidding
 In Internet and Network Economics
, 2010
"... Ads on the Internet are increasingly sold via ad exchanges such as RightMedia, AdECN and Doubleclick Ad Exchange. These exchanges allow realtime bidding, that is, each time the publisher contacts the exchange, the exchange “calls out ” to solicit bids from ad networks. This aspect of soliciting bid ..."
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Cited by 7 (4 self)
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Ads on the Internet are increasingly sold via ad exchanges such as RightMedia, AdECN and Doubleclick Ad Exchange. These exchanges allow realtime bidding, that is, each time the publisher contacts the exchange, the exchange “calls out ” to solicit bids from ad networks. This aspect of soliciting bids introduces a novel aspect, in contrast to existing literature. This suggests developing a joint optimization framework which optimizes over the allocation and well as solicitation. We model this selective call out as an online recurrent Bayesian decision framework with bandwidth type constraints. We obtain natural algorithms with bounded performance guarantees for several natural optimization criteria. We show that these results hold under different call out constraint models, and different arrival processes. Interestingly, the paper shows that under MHR assumptions, the expected revenue of generalized second price auction with reserve is constant factor of the expected welfare. Also the analysis herein allow us prove adaptivity gap type results for the adwords problem. 1
Tight Approximation Algorithms for Maximum Separable Assignment Problems
, 2011
"... A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items int ..."
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Cited by 6 (0 self)
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A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP) 1 and a distributed caching problem (DCP) described in this paper. Given a �approximation algorithm for finding the highest value packing of a single bin, we give (i) A polynomialtime LProunding based ��1 − 1/e���approximation algorithm. (ii) A simple polynomialtime local search ��/� � + 1 � − ��approximation algorithm, for any �> 0. Therefore, for all examples of SAP that admit an approximation scheme for the singlebin problem, we obtain an LPbased algorithm with �1 − 1/e − ��approximation and a local search algorithm with � 1 − ��approximation guarantee. Furthermore, 2 for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LPbased algorithm analysis can be strengthened to give a guarantee of 1 − 1/e. The best previously known approximation algorithm for GAP is a 1approximation by Shmoys and Tardos and Chekuri and Khanna. Our LP algorithm is based on rounding a 2