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Primal-Dual Schemes for Online Matching in Bounded Degree Graphs

Authors Ilan Reuven Cohen , Binghui Peng



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Ilan Reuven Cohen
  • Bar-Ilan University, Ramat Gan, Israel
Binghui Peng
  • Columbia University, New York, NY, USA

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Ilan Reuven Cohen and Binghui Peng. Primal-Dual Schemes for Online Matching in Bounded Degree Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 35:1-35:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.35

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online Matching
  • Primal-dual analysis
  • bounded-degree graph
  • the AdWords problem

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References

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