Online stochastic matching: Beating 1-1/e

Jon Feldman, Aranyak Mehta, Vahab Mirrokni, Shanmugavelayutham Muthukrishnan

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    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 - 1/e barrier. Furthermore, we show that no online algorithm can produce a 1 - ∈ approximation for an arbitrarily small ∈ 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.

    Original languageEnglish (US)
    Title of host publicationProceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009
    Pages117-126
    Number of pages10
    DOIs
    StatePublished - Dec 1 2009
    Event50th Annual Symposium on Foundations of Computer Science, FOCS 2009 - Atlanta, GA, United States
    Duration: Oct 25 2009Oct 27 2009

    Publication series

    NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
    ISSN (Print)0272-5428

    Other

    Other50th Annual Symposium on Foundations of Computer Science, FOCS 2009
    CountryUnited States
    CityAtlanta, GA
    Period10/25/0910/27/09

    Fingerprint

    Display devices
    Flow graphs
    Resource allocation
    Decision making
    Internet

    ASJC Scopus subject areas

    • Computer Science(all)

    Cite this

    Feldman, J., Mehta, A., Mirrokni, V., & Muthukrishnan, S. (2009). Online stochastic matching: Beating 1-1/e. In Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 (pp. 117-126). [5438641] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2009.72

    Online stochastic matching : Beating 1-1/e. / Feldman, Jon; Mehta, Aranyak; Mirrokni, Vahab; Muthukrishnan, Shanmugavelayutham.

    Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009. 2009. p. 117-126 5438641 (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS).

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Feldman, J, Mehta, A, Mirrokni, V & Muthukrishnan, S 2009, Online stochastic matching: Beating 1-1/e. in Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009., 5438641, Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 117-126, 50th Annual Symposium on Foundations of Computer Science, FOCS 2009, Atlanta, GA, United States, 10/25/09. https://doi.org/10.1109/FOCS.2009.72
    Feldman J, Mehta A, Mirrokni V, Muthukrishnan S. Online stochastic matching: Beating 1-1/e. In Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009. 2009. p. 117-126. 5438641. (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2009.72
    Feldman, Jon ; Mehta, Aranyak ; Mirrokni, Vahab ; Muthukrishnan, Shanmugavelayutham. / Online stochastic matching : Beating 1-1/e. Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009. 2009. pp. 117-126 (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS).
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