New algorithms, better bounds, and a novel model for online stochastic matching

Brian Brubach, Karthik Abinav Sankararaman, Aravind Srinivasan, Pan Xu

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

15 Scopus citations


Online matching has received significant attention over the last 15 years due to its close connection to Internet advertising. As the seminal work of Karp, Vazirani, and Vazirani has an optimal (1 - 1/ϵ) competitive ratio in the standard adversarial online model, much effort has gone into developing useful online models that incorporate some stochasticity in the arrival process. One such popular model is the "known I.I.D. model" where different customer-types arrive online from a known distribution. We develop algorithms with improved competitive ratios for some basic variants of this model with integral arrival rates, including: (a) the case of general weighted edges, where we improve the best-known ratio of 0.667 due to Haeupler, Mirrokni and Zadimoghaddam [11] to 0.705; and (b) the vertex-weighted case, where we improve the 0.7250 ratio of Jaillet and Lu [12] to 0.7299. We also consider two extensions, one is "known I.I.D." with non-integral arrival rate and stochastic rewards; the other is "known I.I.D." b-matching with non-integral arrival rate and stochastic rewards. We present a simple non-adaptive algorithm which works well simultaneously on the two extensions. One of the key ingredients of our improvement is the following (offline) approach to bipartite-matching polytopes with additional constraints. We first add several valid constraints in order to get a good fractional solution f; however, these give us less control over the structure of f. We next remove all these additional constraints and randomly move from f to a feasible point on the matching polytope with all coordinates being from the set {0, 1/k,2/k,⋯,1} for a chosen integer k. The structure of this solution is inspired by Jaillet and Lu (Mathematics of Operations Research, 2013) and is a tractable structure for algorithm design and analysis. The appropriate random move preserves many of the removed constraints (approximately [exactly] with high probability [in expectation]). This underlies some of our improvements, and, we hope, could be of independent interest.

Original languageEnglish (US)
Title of host publication24th Annual European Symposium on Algorithms, ESA 2016
EditorsChristos Zaroliagis, Piotr Sankowski
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770156
StatePublished - Aug 1 2016
Externally publishedYes
Event24th Annual European Symposium on Algorithms, ESA 2016 - Aarhus, Denmark
Duration: Aug 22 2016Aug 24 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference24th Annual European Symposium on Algorithms, ESA 2016

All Science Journal Classification (ASJC) codes

  • Software


  • Ad-Allocation
  • Online matching
  • Randomized algorithms


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