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70
The adwords problem: Online keyword matching with budgeted bidders under random permutations
 In Proc. 10th Annual ACM Conference on Electronic Commerge (EC
, 2009
"... We consider the problem of a search engine trying to assign a sequence of search keywords to a set of competing bidders, each with a daily spending limit. The goal is to maximize the revenue generated by these keyword sales, bearing in mind that, as some bidders may eventually exceed their budget, n ..."
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Cited by 70 (6 self)
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We consider the problem of a search engine trying to assign a sequence of search keywords to a set of competing bidders, each with a daily spending limit. The goal is to maximize the revenue generated by these keyword sales, bearing in mind that, as some bidders may eventually exceed their budget, not all keywords should be sold to the highest bidder. We assume that the sequence of keywords (or equivalently, of bids) is revealed online. Our concern will be the competitive ratio for this problem versus the offline optimum. We extend the current literature on this problem by considering the setting where the keywords arrive in a random order. In this setting we are able to achieve a competitive ratio of 1 − ɛ under some mild, but necessary, assumptions.
Sponsored Search Auctions with Markovian Users
"... Abstract. Sponsored search involves running an auction among advertisers who bid in order to have their ad shown next to search results for specific keywords. The most popular auction for sponsored search is the “Generalized Second Price ” (GSP) auction where advertisers are assigned to slots in the ..."
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Cited by 62 (3 self)
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Abstract. Sponsored search involves running an auction among advertisers who bid in order to have their ad shown next to search results for specific keywords. The most popular auction for sponsored search is the “Generalized Second Price ” (GSP) auction where advertisers are assigned to slots in the decreasing order of their score, which is defined as the product of their bid and clickthrough rate. One of the main advantages of this simple ranking is that bidding strategy is intuitive: to move up to a more prominent slot on the results page, bid more. This makes it simple for advertisers to strategize. However this ranking only maximizes efficiency under the assumption that the probability of a user clicking on an ad is independent of the other ads shown on the page. We study a Markovian user model that does not make this assumption. Under this model, the most efficient assignment is no longer a simple ranking function as in GSP. We show that the optimal assignment can be found efficiently (even in nearlinear time). As a result of the more sophisticated structure of the optimal assignment, bidding dynamics become more complex: indeed it is no longer clear that bidding more moves one higher on the page. Our main technical result is that despite the added complexity of the bidding dynamics, the optimal assignment has the property that ad position is still monotone in bid. Thus even in this richer user model, our mechanism retains the core bidding dynamics of the GSP auction that make it useful for advertisers. 1
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
Online bipartite matching with random arrivals: an approach based on strongly factorrevealing lps
 In Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC ’11
, 2011
"... In a seminal paper, Karp, Vazirani, and Vazirani [9] show that a simple ranking algorithm achieves a competitive ratio of 1 − 1/e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1−1/e is also shown to be optimal. Their result also implies that in the r ..."
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Cited by 40 (0 self)
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In a seminal paper, Karp, Vazirani, and Vazirani [9] show that a simple ranking algorithm achieves a competitive ratio of 1 − 1/e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1−1/e is also shown to be optimal. Their result also implies that in the random arrivals model defined by Goel and Mehta [6], where the online nodes arrive in a random order, a simple greedy algorithm achieves a competitive ratio of 1−1/e. In this paper, we study the ranking algorithm in the random arrivals model, and show that it has a competitive ratio of at least 0.696, beating the 1 − 1/e ≈ 0.632 barrier in the adversarial model. Our result also extends to the i.i.d. distribution model of Feldman et al. [5], removing the assumption that the distribution is known. Our analysis has two main steps. First, we exploit certain dominance and monotonicity properties of the ranking algorithm to derive a family of factorrevealing linear programs (LPs). In particular, by symmetry of the ranking algorithm in the random arrivals model, we have the monotonicity property on both sides of the bipartite graph, giving good “strength ” to the LPs. Second, to obtain a good lower bound on the optimal values of all these LPs and hence on the competitive ratio of the algorithm, we introduce the technique of strongly factorrevealing LPs. In particular, we derive a family of modified LPs with similar strength such that the optimal value of any single one of these new LPs is a lower bound on the competitive ratio of the algorithm. This enables us to leverage the power of computer LP solvers to solve for large instances of the new LPs to establish bounds that would otherwise be difficult to attain by human analysis.
Online bipartite matching with unknown distributions
 In STOC
, 2011
"... We consider the online bipartite matching problem in the unknown distribution input model. We show that the Ranking algorithm of [KVV90] achieves a competitive ratio of at least 0.653. This is the first analysis to show an algorithm which breaks the natural 1 − 1/e ‘barrier ’ in the unknown distribu ..."
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Cited by 35 (2 self)
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We consider the online bipartite matching problem in the unknown distribution input model. We show that the Ranking algorithm of [KVV90] achieves a competitive ratio of at least 0.653. This is the first analysis to show an algorithm which breaks the natural 1 − 1/e ‘barrier ’ in the unknown distribution model (our analysis in fact works in the stricter, random order model) and answers an open question in [GM08]. We also describe a family of graphs on which Ranking does no better than 0.727 in the random order model. Finally, we show that for graphs which have k> 1 disjoint perfect matchings, Ranking achieves a competitive ratio of at least 1 −
A Dynamic NearOptimal Algorithm for Online Linear Programming ∗
, 2009
"... A natural optimization model that formulates many online resource allocation and revenue management problems is the online linear program (LP) where the constraint matrix is revealed column by column along with the objective function. We provide a nearoptimal algorithm for this surprisingly general ..."
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Cited by 34 (6 self)
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A natural optimization model that formulates many online resource allocation and revenue management problems is the online linear program (LP) where the constraint matrix is revealed column by column along with the objective function. We provide a nearoptimal algorithm for this surprisingly general class of online problems under the assumption of random order of arrival and some mild conditions on the size of the LP righthandside input. Our learningbased algorithm works by dynamically updating a threshold price vector at geometric time intervals, where the dual prices learned from revealed columns in the previous period are used to determine the sequential decisions in the current period. Our algorithm has a feature of “learning by doing”, and the prices are updated at a carefully chosen pace that is neither too fast nor too slow. In particular, our algorithm doesn’t assume any distribution information on the input itself, thus is robust to data uncertainty and variations due to its dynamic learning capability. Applications of our algorithm include many online multiresource allocation and multiproduct revenue management problems such as online routing and packing, online combinatorial auctions, adwords matching, inventory control and yield management. 1
Near Optimal Online Algorithms and Fast Approximation Algorithms for Resource Allocation Problems
, 2011
"... We present algorithms for a class of resource allocation problems both in the online setting with stochastic input and in the offline setting. This class of problems contains many interesting special cases such as the Adwords problem. In the online setting we introduce a new distributional model cal ..."
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Cited by 31 (3 self)
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We present algorithms for a class of resource allocation problems both in the online setting with stochastic input and in the offline setting. This class of problems contains many interesting special cases such as the Adwords problem. In the online setting we introduce a new distributional model called the adversarial stochastic input model, which is a generalization of the i.i.d model with unknown distributions, where the distributions can change over time. In this model we give a 1 − O(ǫ) approximation algorithm for the resource allocation problem, with almost the weakest possible assumption: the ratio of the maximum amount of resource consumed by any single request to the total capacity of the resource, and the ratio of the profit contributed by any single request to the optimal profit is at most ǫ 2 /log(1/ǫ) 2 where n is the number of resources log n+log(1/ǫ) available. There are instances where this ratio is ǫ 2 /log n such that no randomized algorithm can have a competitive ratio of 1 − o(ǫ) even in the i.i.d model. The upper bound on ratio that we require improves on the previous upperbound for the i.i.d case by a factor of n. Our proof technique also gives a very simple proof that the greedy algorithm has a competitive ratio of 1 −1/e for the Adwords problem in the i.i.d model with unknown distributions, and more generally in the adversarial stochastic input model, when there is no bound on the bid to budget ratio. All the previous proofs assume A full version of this paper, with all the proofs, is available at
When LP is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings (Extended Abstract)
"... Abstract. Consider a random graph model where each possible edge e is present independently with some probability pe. We are given these numbers pe, and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to ..."
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Abstract. Consider a random graph model where each possible edge e is present independently with some probability pe. We are given these numbers pe, and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most ti times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LProunding based constantfactor approximation algorithms for these problems. Our main results are: • We give a 5.75approximation for weighted stochastic matching on general graphs, and a 5approximation on bipartite graphs. This answers an open question from [Chen et al. ICALP 09]. • Combining our LProunding algorithm with the natural greedy algorithm, we give an improved 3.88approximation for unweighted stochastic matching on general graphs and 3.51approximation on bipartite graphs. • We introduce a generalization of the stochastic online matching problem [Feldman et al. FOCS 09] that also models preferenceuncertainty and timeouts of buyers, and give a constant factor approximation algorithm. 1
Online stochastic matching: Online actions based on offline statistics
, 2010
"... We consider the online stochastic matching problem proposed by Feldman et al. [4] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins and the other side represents the set of possible ball types. At each time step, a ball is s ..."
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Cited by 23 (1 self)
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We consider the online stochastic matching problem proposed by Feldman et al. [4] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins and the other side represents the set of possible ball types. At each time step, a ball is sampled independently from the given distribution and it needs to be matched upon its arrival to an empty bin. The goal is to maximize the size of the matching. We present an online algorithm for this problem with a competitive ratio of 0.702. Before our result, algorithms with a competitive ratio better than 1 − 1/e were known under the assumption that the expected number of arriving balls of each type is integral. A key idea of the algorithm is to collect statistics about the decisions of the optimum offline solution using Monte Carlo sampling and use those statistics to guide the decisions of the online algorithm. We also show that no online algorithm can have a competitive ratio better than 0.823. 1
Online vertexweighted bipartite matching and singlebid budgeted allocations
 PROC. OF ACMSIAM SODA
, 2011
"... We study the following vertexweighted online bipartite matching problem: G(U,V,E) is a bipartite graph. The vertices in U have weights and are known ahead of time, while the vertices in V arrive online in an arbitrary order and have to be matched upon arrival. The goal is to maximize the sum of wei ..."
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Cited by 19 (1 self)
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We study the following vertexweighted online bipartite matching problem: G(U,V,E) is a bipartite graph. The vertices in U have weights and are known ahead of time, while the vertices in V arrive online in an arbitrary order and have to be matched upon arrival. The goal is to maximize the sum of weights of the matched vertices in U. When all the weights are equal, this reduces to the classic online bipartite matching problem for which Karp, Vazirani and Vazirani gave an optimal ( 1 − 1 ecompetitive algorithm in their seminal work [KVV90]. Our main result is an optimal ( 1 − 1 ecompetitive randomized algorithm for general vertex weights. We use random perturbations of weights by appropriately chosen multiplicative factors. Our solution constitutes the first known generalization of the algorithm in [KVV90] in this model and provides new insights into the role of randomization in online allocation problems. It also effectively solves the problem of online budgeted allocations [MSVV05] in the case when an agent makes the same bid for any desired item, even if the bid is comparable to his budgetcomplementing the results of [MSVV05, BJN07] which apply when the bids are much smaller than the budgets.