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Online Mixed Ring Covering Problem with Two Nodes

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Abstract

In this paper, we study the online mixed ring covering problem, where the ring contains two nodes and undirected and bidirected links. A sequence of flows arrives one by one, where each flow has a traffic demand for each pair of nodes in the ring. The objective is to maximize the minimum load of the ring link, where the load of a link is the total demand of the flows sent to that link. We consider the problem in three different scenarios: splittable, integer splittable and unsplittable. When the demands are splittable, we present an optimal online algorithm with a competitive ratio that is no more than \(\frac{4}{3}\). When the demands are integer splittable, we present an optimal online algorithm with a competitive ratio that is no more than 2. When the demands are unsplittable, we prove that the lower bound for this case is \(\infty\), and few researchers have provided this result. Then, we consider a special case of the online mixed ring covering problem when the demands are unsplittable, which has a buffer size of K, where K is the number of flows temporarily stored in the buffer. We prove that the competitive ratio for any positive integer K is at least 2. For \(K=1\), we present an online algorithm with a competitive ratio that is no more than 3. For \(K=2\), we present an online algorithm with a competitive ratio that is no more than \(\frac{3+\sqrt{5}}{2}\approx 2.618\).

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Funding

The work is supported by Weidong Li of part by the Program for Excellent Young Talents of Yunnan University, Training Program of National Science Fund for Distinguished Young Scholars, IRTSTYN, and Key Joint Project of the Science and Technology Department of Yunnan Province and Yunnan University (No. 2018FY001(-014)).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yunnan University, Kunming, 650504, China

    Man Xiao & Weidong Li

  2. School of Information Science and Engineering, Yunnan University, Kunming, 650504, China

    Xiaofei Liu

Authors
  1. Man Xiao
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  2. Weidong Li
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  3. Xiaofei Liu
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Contributions

Conceptualization: X.M. and W.L.; methodology: X.M., W.L. and X.L.; validation: X.M., W.L. and X.L.; formal analysis: X.M. and X.L.; investigation: W.L.; resources” W.L.; writing—original draft preparation: X.M.; writing—review and editing: W.L. and X.L.; supervision: W.L. and X.L.; project administration: W.L. and X.L.; funding acquisition: W.L. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Xiaofei Liu.

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Xiao, M., Li, W. & Liu, X. Online Mixed Ring Covering Problem with Two Nodes. Oper. Res. Forum 4, 8 (2023). https://doi.org/10.1007/s43069-022-00189-x

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  • DOI: https://doi.org/10.1007/s43069-022-00189-x

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