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An \(O(\sqrt{k})\)-Approximation Algorithm for Minimum Power k Edge Disjoint st-Paths

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Unity of Logic and Computation (CiE 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13967))

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Abstract

In minimum power network design problems we are given an undirected graph \(G=(V,E)\) with edge costs \(\{c_e:e \in E\}\). The goal is to find an edge set \(F \subseteq E\) that satisfies a prescribed property of minimum power \(p_c(F)=\sum _{v \in V} \max \{c_e: e \in F \text{ is } \text{ incident } \text{ to } v\}\). In the Min-Power k Edge Disjoint st -Paths problem F should contain k edge disjoint st-paths. The problem admits a k-approximation algorithm, and it was an open question whether it admits an approximation ratio sublinear in k even for unit costs. We give a \(4\sqrt{2k}\)-approximation algorithm for general costs.

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Notes

  1. 1.

    In [1] it is also claimed that Min-Power 2 Disjoint Paths admits a 1.5-approximation algorithm, but the proof was found to contain an error [5]. Obtaining an approximation ratio better than 2 for Min-Power 2 Edge/Node Disjoint Paths is still a major open problem in the field.

  2. 2.

    To see this, assume w.l.o.g. that \(|B| \ge |A|\) and note that \(j=|A \, \backslash B| \ge 1\). Then \(|A \cap B|^2+|A \cup B|^2= (|A|-j)^2+(|B|+j)^2= |A|^2+|B|^2+2j(|B|-|A|+j) >|A|^2+|B|^2\).

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Acknowledgment

I thank anonymous referees for useful comments that helped to improve the presentation of this paper.

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Authors and Affiliations

  1. The Open University of Israel, Ra’anana, Israel

    Zeev Nutov

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  1. Zeev Nutov
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Correspondence to Zeev Nutov .

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Editors and Affiliations

  1. University of Milano-Bicocca, Milan, Italy

    Gianluca Della Vedova

  2. Kutaisi International University, Kutaisi, Georgia

    Besik Dundua

  3. University of Wisconsin, Madison, WI, USA

    Steffen Lempp

  4. University of Göttingen, Göttingen, Germany

    Florin Manea

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Nutov, Z. (2023). An \(O(\sqrt{k})\)-Approximation Algorithm for Minimum Power k Edge Disjoint st-Paths. In: Della Vedova, G., Dundua, B., Lempp, S., Manea, F. (eds) Unity of Logic and Computation. CiE 2023. Lecture Notes in Computer Science, vol 13967. Springer, Cham. https://doi.org/10.1007/978-3-031-36978-0_23

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  • DOI: https://doi.org/10.1007/978-3-031-36978-0_23

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-36977-3

  • Online ISBN: 978-3-031-36978-0

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