## Online stochastic matching: Online actions based on offline statistics (2010)

Citations: | 17 - 1 self |

### BibTeX

@MISC{Manshadi10onlinestochastic,

author = {Vahideh H. Manshadi and Shayan Oveis and Gharan Amin Saberi},

title = {Online stochastic matching: Online actions based on offline statistics},

year = {2010}

}

### OpenURL

### Abstract

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

### Citations

143 |
An optimal algorithm for on-line bipartite matching
- Karp, Vazirani, et al.
- 1990
(Show Context)
Citation Context ...matched to atmostone ball. The goal is to maximize the expected number of non-empty bins at time b. When the balls are chosen by an adversary instead of a random process, Karp, Vazirani, and Vazirani =-=[8]-=- gave a simple and elegant randomized algorithm that achievesacompetitive ratioof1−1/e. We present the first algorithm for this problem that im∗ Department of ElectricalEngineering, Stanford Universit... |

136 | Cuckoo Hashing
- Pagh, Rodler
- 2004
(Show Context)
Citation Context ...optimum solution. The optimum offline algorithm will essentially find the maximum matching between all arrived ball types and the bins. The size of this maximum matching is studied by Path and Rodler =-=[10]-=-. There, the problem is defined as follows: there are b keys to be hashed into n buckets, each capable of holding a single key. Each key has k ≥ 2 (distinct) associated buckets chosen uniformly at ran... |

106 | Adwords and generalized on-line matching
- Mehta, Saberi, et al.
- 2007
(Show Context)
Citation Context ...hat for d-regular graphs, a simple randomized algorithm achieves a competitive ratio of 1 − O(1/ √ d) [1]. A close line of work to the online matching is the online b-matching and the AdWords problem =-=[9, 2]-=-. Mehtaetal. [9]developeda(1−e −1 )onlinealgorithm in the adversarial case. Recently, Devanur and Hayes improved the competitive ratio to (1 − ɛ) in the stochastic case where the sequence of arrivals ... |

41 | The adwords problem: Online keyword matching with budgeted bidders under random permutations
- Devanur, Hayes
- 2009
(Show Context)
Citation Context ...hat for d-regular graphs, a simple randomized algorithm achieves a competitive ratio of 1 − O(1/ √ d) [1]. A close line of work to the online matching is the online b-matching and the AdWords problem =-=[9, 2]-=-. Mehtaetal. [9]developeda(1−e −1 )onlinealgorithm in the adversarial case. Recently, Devanur and Hayes improved the competitive ratio to (1 − ɛ) in the stochastic case where the sequence of arrivals ... |

32 |
Online stochastic matching: Beating 1 - 1/e
- Feldman, Mehta, et al.
- 2009
(Show Context)
Citation Context ...c Matching: Online Actions Based on Offline Statistics Vahideh H. Manshadi ∗ Shayan Oveis Gharan † Amin Saberi † Abstract 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 tim... |

20 | Tight thresholds for cuckoo hashing via XORSAT
- Dietzfelbinger, Goerdt, et al.
(Show Context)
Citation Context ... the competitive ratio of any deterministic or randomized online algorithm. For analyzing this example, we use the expected size of a maximum matching of a random bipartite graph recently computed by =-=[3, 6, 5]-=- in the context of Random SAT and cuckoo hashing. 1.1 Related Work Bipartite matching problems arecentralin algorithmsand combinatorialoptimization and arise naturally in several applications such as ... |

13 | Improved bounds for online stochastic matching
- Bahmani, Kapralov
- 2010
(Show Context)
Citation Context ..., Stanford University, Stanford, CA 94305. Email:{shayan,saberi}@stanford.edu. provesthe 1−1/ecompetitive ratio for the stochastic versionin its generalform. Previously, Feldman et al. [4] (and later =-=[1]-=-) used a very interesting combinatorial algorithm to show that this is possible when the arrival rate of every ball, that is the expected number of times it appears, is integral. This assumption, even... |

9 | Maximum Matchings in Random Bipartite Graphs and the Space Utilization of Cuckoo Hashtables, arXiv:0910.5535v3 [cs.DS] 30 M. Karonski and B. Pittel, Existence of a perfect matching in a random (1+e −1 )–out bipartite graph
- Frieze, Melsted
- 2003
(Show Context)
Citation Context ... the competitive ratio of any deterministic or randomized online algorithm. For analyzing this example, we use the expected size of a maximum matching of a random bipartite graph recently computed by =-=[3, 6, 5]-=- in the context of Random SAT and cuckoo hashing. 1.1 Related Work Bipartite matching problems arecentralin algorithmsand combinatorialoptimization and arise naturally in several applications such as ... |

2 |
Sharp load thresholds for cuckoo hashing. arXiv report 0910.5147v1, 2009
- Fountoulakis, Panagiotou
- 2005
(Show Context)
Citation Context ...eralonlinealgorithms,underarbitraryrates. In this example, we use calculations on the size of perfect matchingsin randombipartite graphsstudied earlier in the context of Random SAT and cuckoo hashing =-=[3, 6, 5]-=-. For a set Z of bins, define Yk to be a set of (|Z| k ) vertices, each connected to a distinct subset of cardinality k of Z. These sets will play an important role in constructing examples with large... |

1 |
random input models with applications to adwords
- Onlinebudgetedmatchingin
- 2008
(Show Context)
Citation Context ...a ball arrives it reveals its incident edges. They proved that a simple randomized on-line algorithm achieves (1−e −1 ) and this factor is the best possible performance. More recently, Goel and Mehta =-=[7]-=- and Feldman et al. [4] studied the problem under stochastic assumptions. Goel and Mehta [7] assumed the graph is unknown but the sequence of arrivals is a random permutation. They showed that a greed... |