Primal-Dual Schemes for Online Matching in Bounded Degree Graphs

Ilan Reuven Cohen; Binghui Peng

doi:10.4230/LIPIcs.ESA.2023.35

Primal-Dual Schemes for Online Matching in Bounded Degree Graphs

Ilan Reuven Cohen, Binghui Peng

Bar-Ilan University - The Alexander Kofkin Faculty of Engineering

Columbia University

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

Abstract

We explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem [8], the allocation problem [5], and the AdWords problem [13] in a beyond-worst-case setting. Specifically, we assume that G is a (k, d)-bounded degree graph, introduced by Naor and Wajc [14]. 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 [6, 3] 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.

Original language	English
Title of host publication	31st Annual European Symposium on Algorithms, ESA 2023
Editors	Inge Li Gortz, Martin Farach-Colton, Simon J. Puglisi, Grzegorz Herman
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959772952
DOIs	https://doi.org/10.4230/LIPIcs.ESA.2023.35
State	Published - Sep 2023
Event	31st Annual European Symposium on Algorithms, ESA 2023 - Amsterdam, Netherlands Duration: 4 Sep 2023 → 6 Sep 2023

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	274
ISSN (Print)	1868-8969

Conference

Conference	31st Annual European Symposium on Algorithms, ESA 2023
Country/Territory	Netherlands
City	Amsterdam
Period	4/09/23 → 6/09/23

Bibliographical note

Publisher Copyright:
© Ilan Reuven Cohen and Binghui Peng.

Funding

Funding Ilan Reuven Cohen: Supported by the Israel Science Foundation grant No. 1737/21. Binghui Peng: Supported by NSF IIS-1838154, CCF-2106429, CCF-2107187, CCF-1763970, CCF-2212233.

Funders	Funder number
National Science Foundation	CCF-2107187, CCF-1763970, CCF-2212233, IIS-1838154, CCF-2106429
Israel Science Foundation	1737/21

Keywords

Online Matching
Primal-dual analysis
bounded-degree graph
the AdWords problem

Access to Document

10.4230/LIPIcs.ESA.2023.35

Cite this

Cohen, I. R., & Peng, B. (2023). Primal-Dual Schemes for Online Matching in Bounded Degree Graphs. In I. Li Gortz, M. Farach-Colton, S. J. Puglisi, & G. Herman (Eds.), 31st Annual European Symposium on Algorithms, ESA 2023 Article 35 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 274). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ESA.2023.35

Cohen, Ilan Reuven ; Peng, Binghui. / Primal-Dual Schemes for Online Matching in Bounded Degree Graphs. 31st Annual European Symposium on Algorithms, ESA 2023. editor / Inge Li Gortz ; Martin Farach-Colton ; Simon J. Puglisi ; Grzegorz Herman. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2023. (Leibniz International Proceedings in Informatics, LIPIcs).

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abstract = "We explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem [8], the allocation problem [5], and the AdWords problem [13] in a beyond-worst-case setting. Specifically, we assume that G is a (k, d)-bounded degree graph, introduced by Naor and Wajc [14]. 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 [6, 3] 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.",

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Cohen, IR & Peng, B 2023, Primal-Dual Schemes for Online Matching in Bounded Degree Graphs. in I Li Gortz, M Farach-Colton, SJ Puglisi & G Herman (eds), 31st Annual European Symposium on Algorithms, ESA 2023., 35, Leibniz International Proceedings in Informatics, LIPIcs, vol. 274, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 31st Annual European Symposium on Algorithms, ESA 2023, Amsterdam, Netherlands, 4/09/23. https://doi.org/10.4230/LIPIcs.ESA.2023.35

Primal-Dual Schemes for Online Matching in Bounded Degree Graphs. / Cohen, Ilan Reuven; Peng, Binghui.
31st Annual European Symposium on Algorithms, ESA 2023. ed. / Inge Li Gortz; Martin Farach-Colton; Simon J. Puglisi; Grzegorz Herman. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2023. 35 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 274).

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

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Cohen IR, Peng B. Primal-Dual Schemes for Online Matching in Bounded Degree Graphs. In Li Gortz I, Farach-Colton M, Puglisi SJ, Herman G, editors, 31st Annual European Symposium on Algorithms, ESA 2023. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2023. 35. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.ESA.2023.35

Primal-Dual Schemes for Online Matching in Bounded Degree Graphs

Abstract

Publication series

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Bibliographical note

Funding

Keywords

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