License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ESA.2023.35
URN: urn:nbn:de:0030-drops-186884
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18688/
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Cohen, Ilan Reuven ; Peng, Binghui

Primal-Dual Schemes for Online Matching in Bounded Degree Graphs

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Abstract

We explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem [Kalyanasundaram and Pruhs, 2000], the allocation problem [Buchbinder et al., 2007], and the AdWords problem [Mehta et al., 2007] in a beyond-worst-case setting. Specifically, we assume that G is a (k, d)-bounded degree graph, introduced by Naor and Wajc [Naor and Wajc, 2018]. Such graphs model natural properties on the degrees of advertisers and queries in the allocation and AdWords problems. While previous work only considers the scenario where k ≥ d, we consider the interesting intermediate regime of k ≤ d and prove a tight competitive ratio as a function of k,d (under the small-bid assumption) of τ(k,d) = 1 - (1-k/d)⋅(1-1/d)^{d - k} for the b-matching and allocation problems. We exploit primal-dual schemes [Buchbinder et al., 2009; Azar et al., 2017] to design and analyze the corresponding tight upper and lower bounds. Finally, we show a separation between the allocation and AdWords problems. We demonstrate that τ(k,d) competitiveness is impossible for the AdWords problem even in (k,d)-bounded degree graphs.

BibTeX - Entry

@InProceedings{cohen_et_al:LIPIcs.ESA.2023.35,
  author =	{Cohen, Ilan Reuven and Peng, Binghui},
  title =	{{Primal-Dual Schemes for Online Matching in Bounded Degree Graphs}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{35:1--35:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18688},
  URN =		{urn:nbn:de:0030-drops-186884},
  doi =		{10.4230/LIPIcs.ESA.2023.35},
  annote =	{Keywords: Online Matching, Primal-dual analysis, bounded-degree graph, the AdWords problem}
}

Keywords: Online Matching, Primal-dual analysis, bounded-degree graph, the AdWords problem
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

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