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On PTAS for the Geometric Maximum Connected k-Factor Problem

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Optimization and Applications (OPTIMA 2019)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1145))

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Abstract

We consider the Connected k-factor problem (k-CFP): given a complete edge-weighted n-vertex graph, the goal is to find a connected k-regular spanning subgraph of maximum or minimum total weight. The problem is called geometric, if the vertices of a graph correspond to a set of points in a normed space \(\mathbb R^d\) and the weight of an edge is the distance between its endpoints. The k-CFP is a natural generalization of the well-known Traveling Salesman Problem, which is equivalent to the 2-CFP. In this paper we complement the known \((1-1/ k^2)\)-approximation algorithm for the maximum k-CFP from [Baburin et al., 2007] with an approximation algorithm for the geometric k-CFP, that guarantees a relative error \(\varepsilon = O\left( (k/n)^{1/(d+2)}\right) \). Together these two algorithms form an asymptotically optimal algorithm for the geometric k-CFP with an arbitrary value of k in an arbitrary normed space of fixed dimension d. Finally, the asymptotically optimal algorithm can be easily transformed into a PTAS for the considered geometric problem.

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Funding

The authors are supported by the program of fundamental scientific researches of the SB RAS, project 0314-2019-0014 and by the Ministry of Science and Higher Education of the Russian Federation under the 5-100 Excellence Programme.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia

    Edward Gimadi, Ivan Rykov & Oxana Tsidulko

  2. Department of Mechanics and Mathematics, Novosibirsk State University, 1 Pirogova Street, 630090, Novosibirsk, Russia

    Edward Gimadi, Ivan Rykov & Oxana Tsidulko

Authors
  1. Edward Gimadi
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  2. Ivan Rykov
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  3. Oxana Tsidulko
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Corresponding author

Correspondence to Oxana Tsidulko .

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Editors and Affiliations

  1. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  2. Krasovskii Institute of Mathematics and Mechanics, Yekaterinburg, Russia

    Michael Khachay

  3. FRC CSC RAS, Moscow, Russia

    Vlasta Malkova

  4. FRC CSC RAS, Moscow, Russia

    Mikhail Posypkin

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Gimadi, E., Rykov, I., Tsidulko, O. (2020). On PTAS for the Geometric Maximum Connected k-Factor Problem. In: Jaćimović, M., Khachay, M., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2019. Communications in Computer and Information Science, vol 1145. Springer, Cham. https://doi.org/10.1007/978-3-030-38603-0_15

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  • DOI: https://doi.org/10.1007/978-3-030-38603-0_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-38602-3

  • Online ISBN: 978-3-030-38603-0

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