### 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 language | English (US) |
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Title of host publication | Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 |

Pages | 117-126 |

Number of pages | 10 |

DOIs | |

State | Published - Dec 1 2009 |

Event | 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 - Atlanta, GA, United States Duration: Oct 25 2009 → Oct 27 2009 |

### Publication series

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

### Other

Other | 50th Annual Symposium on Foundations of Computer Science, FOCS 2009 |
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Country | United States |

City | Atlanta, GA |

Period | 10/25/09 → 10/27/09 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Online stochastic matching

T2 - Beating 1-1/e

AU - Feldman, Jon

AU - Mehta, Aranyak

AU - Mirrokni, Vahab

AU - Muthukrishnan, Shanmugavelayutham

PY - 2009/12/1

Y1 - 2009/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77952348885&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952348885&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2009.72

DO - 10.1109/FOCS.2009.72

M3 - Conference contribution

AN - SCOPUS:77952348885

SN - 9780769538501

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 117

EP - 126

BT - Proceedings - 50th Annual Symposium on Foundations of Computer Science, FOCS 2009

ER -