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34
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.
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 33 (5 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
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 22 (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
Optimal Auctions with Positive Network Externalities
"... We consider the problem of designing auctions in social networks for goods that exhibit singleparameter submodular network externalities in which a bidder’s value for an outcome is a fixed private type times a known submodular function of the allocation of his friends. Externalities pose many issue ..."
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Cited by 16 (3 self)
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We consider the problem of designing auctions in social networks for goods that exhibit singleparameter submodular network externalities in which a bidder’s value for an outcome is a fixed private type times a known submodular function of the allocation of his friends. Externalities pose many issues that are hard to address with traditional techniques; our work shows how to resolve these issues in a specific setting of particular interest. We operate in a Bayesian environment and so assume private values are drawn according to known distributions. We prove that the optimal auction is APXhard. Thus we instead design auctions whose revenue approximates that of the optimal auction. Our main result considers stepfunction externalities in which a bidder’s value for an outcome is either zero, or equal to his private type if at least one friend has the good. For these e e+1 settings, we provide aapproximation. We also give a 0.25approximation auction for general singleparameter submodular network externalities, and discuss optimizing over a class of simple pricing strategies.
Better bounds for matchings in the streaming model
 In SODA
, 2013
"... In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input when Õ(n) space is allowed, where n is the number of vertices in t ..."
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Cited by 10 (3 self)
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In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input when Õ(n) space is allowed, where n is the number of vertices in the input graph. Two natural variants of this problem have been considered in the literature: (1) the edge arrival setting, where edges arrive in the stream and (2) the vertex arrival setting, where vertices on one side of the graph arrive in the stream together with all their incident edges. The latter setting has also been studied extensively in the context of online algorithms, where each arriving vertex has to either be matched irrevocably or discarded upon arrival. In the online setting, the celebrated algorithm of KarpVaziraniVazirani achieves a 1 − 1/e approximation by crucially using randomization (and using Õ(n) space). Despite the fact that the streaming model is less restrictive in that the algorithm is not constrained to match vertices irrevocably upon arrival, the best known approximation in the streaming model with vertex arrivals and Õ(n) space is the same factor of 1 − 1/e. We show that no (possibly randomized) single pass streaming algorithm constrained to use Õ(n) space can achieve a better than 1 − 1/e approximation to maximum matching, even in the vertex arrival setting.
Maximum matching in semistreaming with few passes
 CoRR
"... We present three semistreaming algorithms for Maximum Bipartite Matching with one and two passes. Our onepass semistreaming algorithm is deterministic and returns a matching of size at least 1/2 + 0.005 times the optimal matching size in expectation, assuming that edges arrive one by one in (unif ..."
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We present three semistreaming algorithms for Maximum Bipartite Matching with one and two passes. Our onepass semistreaming algorithm is deterministic and returns a matching of size at least 1/2 + 0.005 times the optimal matching size in expectation, assuming that edges arrive one by one in (uniform) random order. Our first twopass algorithm is randomized and returns a matching of size at least 1/2 + 0.019 times the optimal matching size in expectation (over its internal random coin flips) for any arrival order. These two algorithms apply the simple Greedy matching algorithm several times on carefully chosen subgraphs as a subroutine. Furthermore, we present a twopass deterministic algorithm for any arrival order returning a matching of size at least 1/2 + 0.019 times the optimal matching size. This algorithm is built on ideas from the computation of semimatchings. 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 6 (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
Asymptotically Optimal Algorithm for Stochastic Adwords
, 2012
"... In this paper we consider the adwords problem in the unknown distribution model. We consider the case where the budget to bid ratio k is at least 2, and give improved competitive ratios. Earlier results had competitive ratios better than 1 − 1/e only for “large enough ” k, while our competitive rati ..."
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Cited by 5 (0 self)
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In this paper we consider the adwords problem in the unknown distribution model. We consider the case where the budget to bid ratio k is at least 2, and give improved competitive ratios. Earlier results had competitive ratios better than 1 − 1/e only for “large enough ” k, while our competitive ratio increases continuously with k. For k = 2 the competitive ratio we get is 0.729 and it is 0.9 for k = 16. We also improve the asymptotic competitive ratio for large k from 1 − O ( p log n/k) to 1 − O ( p 1/k), thus removing any dependence on n, the number of advertisers. This ratio is optimal, even with known distributions. That is, even if an algorithm is tailored to the distribution, it cannot get a competitive ratio of 1 − o ( p 1/k), whereas our algorithm does not depend on the distribution. The algorithm is rather simple, it computes a score for every advertiser based on his original budget, the remaining budget and the remaining number of steps in the algorithm and assigns a query to the advertiser with the highest bid plus his score. The analysis is based on a “hybrid argument ” that considers algorithms that are part actual, part hypothetical, to prove that our (actual) algorithm is better than a completely hypothetical algorithm whose performance is easy to analyze.
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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Cited by 5 (1 self)
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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.