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2009 50th Annual IEEE Symposium on Foundations of Computer Science (2009)
Atlanta, GA
Oct. 25, 2009 to Oct. 27, 2009
ISSN: 0272-5428
ISBN: 978-1-4244-5116-6
pp: 117-126
We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of 1- 1/e ¿ 0.632, a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the 1 - 1/e bound. Our main result is a 0.67-approximation online algorithm for stochastic bipartite matching, breaking this 1 - ¿ barrier. Furthermore, we show that no online algorithm can produce a 1 - ¿ approximation for an arbitrarily small e for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-)matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.
decision making, graph theory, greedy algorithms, Internet, resource allocation, stochastic processes

J. Feldman, A. Mehta, V. Mirrokni and S. Muthukrishnan, "Online Stochastic Matching: Beating 1-1/e," 2009 50th Annual IEEE Symposium on Foundations of Computer Science(FOCS), Atlanta, Georgia, 2010, pp. 117-126.
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