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39
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
Online Task Assignment in Crowdsourcing Markets
"... We explore the problem of assigning heterogeneous tasks to workers with different, unknown skill sets in crowdsourcing markets such as Amazon Mechanical Turk. We first formalize the online task assignment problem, in which a requester has a fixed set of tasks and a budget that specifies how many tim ..."
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Cited by 32 (3 self)
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We explore the problem of assigning heterogeneous tasks to workers with different, unknown skill sets in crowdsourcing markets such as Amazon Mechanical Turk. We first formalize the online task assignment problem, in which a requester has a fixed set of tasks and a budget that specifies how many times he would like each task completed. Workers arrive one at a time (with the same worker potentially arriving multiple times), and must be assigned to a task upon arrival. The goal is to allocate workers to tasks in a way that maximizes the total benefit that the requester obtains from the completed work. Inspired by recent research on the online adwords problem, we present a twophase explorationexploitation assignment algorithm and prove that it is competitive with respect to the optimal offline algorithm which has access to the unknown skill levels of each worker. We empirically evaluate this algorithm using data collected on Mechanical Turk and show that it performs better than random assignment or greedy algorithms. To our knowledge, this is the first work to extend the online primaldual technique used in the online adwords problem to a scenario with unknown parameters, and the first to offer an empirical validation of an online primaldual algorithm.
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
Geometry of online packing linear programs
"... Abstract. We consider packing LP’s with m rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive to obtain a feasible solution maximizing the expe ..."
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Cited by 8 (1 self)
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Abstract. We consider packing LP’s with m rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive to obtain a feasible solution maximizing the expected reward. Previous (1 − )competitive algorithms require the righthand side of the LP to be
Online Allocation of Display Ads with Smooth Delivery
"... Display ads on the Internet are often sold in bundles of thousands or millions of impressions over a particular time period, typically weeks or months. Ad serving systems that assign ads to pages on behalf of publishers must satisfy these contracts, but at the same time try to maximize overall quali ..."
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Cited by 8 (0 self)
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Display ads on the Internet are often sold in bundles of thousands or millions of impressions over a particular time period, typically weeks or months. Ad serving systems that assign ads to pages on behalf of publishers must satisfy these contracts, but at the same time try to maximize overall quality of placement. This is usually modeled in the literature as an online allocation problem, where contracts are represented by overall delivery constraints over a finite time horizon. However this model misses an important aspect of ad delivery: time homogeneity. Advertisers who buy these packages expect their ad to be shown smoothly throughout the purchased time period, in order to reach a wider audience, to have a sustained impact, and to support the ads they are running on other media (e.g., television). In this paper we formalize this problem using several nested packing constraints, and develop a tight (1−1/e)competitive online algorithm for this problem. Our algorithms and analysis require novel techniques as they involve online computation of multiple dual variables per ad. We then show the effectiveness of our algorithms through exhaustive simulation studies on real data sets. 1.
Simultaneous approximations for adversarial and stochastic online budgeted allocation problems
 In SODA
, 2012
"... Motivated by online ad allocation, we study the problem of simultaneous approximations for the adversarial and stochastic online budgeted allocation problem. This problem consists of a bipartite graph G = (X, Y, E), where the nodes of Y along with their corresponding capacities are known beforehand ..."
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Cited by 7 (1 self)
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Motivated by online ad allocation, we study the problem of simultaneous approximations for the adversarial and stochastic online budgeted allocation problem. This problem consists of a bipartite graph G = (X, Y, E), where the nodes of Y along with their corresponding capacities are known beforehand to the algorithm, and the nodes of X arrive online. When a node of X arrives, its incident edges, and their respective weights are revealed, and the algorithm can match it to a neighbor in Y. The objective is to maximize the weight of the final matching, while respecting the capacities. When nodes arrive in an adversarial order, the best competitive ratio is known to be 1 − 1/e, and it can be achieved by the Ranking [18], and its generalizations (Balance [16, 21]). On the other hand, if the nodes arrive through a random permutation, it is possible to achieve a competitive ratio of 1 − ɛ [9]. In this paper we design algorithms that achieve a competitive ratio better than 1 − 1/e on average, while preserving a nearly optimal worst case competitive ratio. Ideally, we want to achieve the best of both worlds, i.e, to design an algorithm with the optimal competitive ratio in both the adversarial and random arrival models. We achieve this for unweighted graphs, but show that it is not possible for weighted graphs. In particular, for unweighted graphs, under some mild assumptions, we show that Balance achieves a competitive ratio of 1 − ɛ in a random permutation model. For weighted graphs, however, we prove this is not possible; we prove that no online algorithm that achieves an approximation factor of 1 − 1 e for the worstcase inputs may achieve an average approximation factor better than 97.6 % for random inputs. In light of this hardness result, we aim to design algorithms with improved approximation ratios in the random arrival in the worst case. To this end, we show the algorithm proposed by [21] achieves a competitive ratio of 0.76 for the random ratio in the worst case. model while preserving the competitive ratio of 1 − 1
Online MaketoOrder Joint Replenishment Model: Primal DualCompetitive Algorithms
"... Abstract In this paper, we study an online maketoorder variant of the classical joint replenishment problem(JRP) that has been studied extensively over the years and plays a fundamental role in broader planning issues, such as the management of supply chains. In contrast to the traditional approac ..."
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Cited by 7 (2 self)
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Abstract In this paper, we study an online maketoorder variant of the classical joint replenishment problem(JRP) that has been studied extensively over the years and plays a fundamental role in broader planning issues, such as the management of supply chains. In contrast to the traditional approaches of the stochastic inventory theory, we study the problem using competitive analysis against a worstcase adversary. Our main result is a 3competitive deterministic algorithm for the online version of the JRP. We alsoprove a lower bound of approximately 2.64 on the competitiveness of any deterministic online algorithmfor the problem. Our algorithm is based on a novel primaldual approach using a new linear programming relaxation of the offline JRP model. The primaldual approach that we propose departs from previousprimaldual and online algorithms in rather significant ways. We believe that this approach can extend the range of problems to which online and primaldual algorithms can be applied and analyzed.
Yield Optimization of Display Advertising with Ad Exchange
, 2011
"... Abstract. In light of the growing market of Ad Exchanges for the realtime sale of advertising slots, publishers face new challenges in choosing between the allocation of contractbased reservation ads and spot market ads. In this setting, the publisher should take into account the tradeoff between ..."
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Abstract. In light of the growing market of Ad Exchanges for the realtime sale of advertising slots, publishers face new challenges in choosing between the allocation of contractbased reservation ads and spot market ads. In this setting, the publisher should take into account the tradeoff between shortterm revenue from an Ad Exchange and the longterm benefits of delivering good quality spots to the reservation ads. In this paper, we formalize this combined optimization problem as a stochastic control problem and derive an efficient policy for online ad allocation in settings with general joint distribution over placement quality and exchange bids. We prove asymptotic optimality of this policy in terms of any tradeoff between quality of delivered reservation ads and revenue from the exchange, and provide a rigorous bound for its convergence rate to the optimal policy. We also give experimental results on data derived from real publisher inventory, showing that our policy can achieve any Paretooptimal point on the quality vs. revenue curve. Finally, we study a parametric trainingbased algorithm in which instead of learning the dual variables from a data sample (as is done in nonparametric trainingbased algorithms), we learn the parameters of the distribution and construct those dual variables from the learned parameter values. We compare parametric and nonparametric ways to estimate from data both analytically and experimentally in the special case without the ad exchange, and show that though both methods converge to the optimal policy as the sample size grows, our parametric method converges faster, and thus performs better on smaller samples.
An Expressive Mechanism for Auctions on the Web
 WWW 2011
, 2011
"... Auctions are widely used on the Web. Applications range from internet advertising to platforms such as eBay. In most of these applications the auctions in use are single/multiitem auctions with unit demand. The main drawback of standard mechanisms for this type of auctions, such as VCG and GSP, is t ..."
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Cited by 5 (0 self)
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Auctions are widely used on the Web. Applications range from internet advertising to platforms such as eBay. In most of these applications the auctions in use are single/multiitem auctions with unit demand. The main drawback of standard mechanisms for this type of auctions, such as VCG and GSP, is the limited expressiveness that they offer to the bidders. The General Auction Mechanism (GAM) of [1] is taking a first step towards addressing the problem of limited expressivenessbycomputingabidderoptimal, envyfree outcome for linear utility functions with identical slopes and a single discontinuity per bidderitem pair. We show that in many practical situations this does not suffice to adequately model the preferences of the bidders, and we overcome this problem bypresentingthefirst mechanism for piecewise linear