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On \((1+\varepsilon )\)-approximate Data Reduction for the Rural Postman Problem

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Book cover Mathematical Optimization Theory and Operations Research (MOTOR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11548))

Abstract

Given a graph \(G=(V,E)\) with edge weights and a subset \(R\subseteq E\) of required edges, the NP-hard Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. The number b of vertices incident to an odd number of edges of R and the number c of connected components formed by the edges in R are both bounded from above by the number of edges that has to be traversed additionally to the required ones. We show how to reduce any RPP instance I to an RPP instance \(I'\) with \(2b+O(c/\varepsilon )\) vertices in \(O(n^3)\) time so that any \(\alpha \)-approximate solution for \(I'\) gives an \(\alpha (1+\varepsilon )\)-approximate solution for I, for any \(\alpha \ge 1\) and \(\varepsilon >0\). That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We make first steps towards a PSAKS with respect to the parameter c.

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Notes

  1. 1.

    All omitted proofs can be found in the full version of this paper, available on arXiv: https://arxiv.org/abs/1812.10131.

  2. 2.

    That is, \(T_i\) is the folklore 2-approximation of a Steiner tree with terminals \(B_i\) in \(G\langle R_i\rangle \).

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Acknowledgments

René van Bevern and Oxana Yu. Tsidulko are supported by the Russian Foundation for Basic Research, project 18-501-12031 NNIO_a, and by the Ministry of Science and Higher Education of the Russian Federation under the 5-100 Excellence Programme. Till Fluschnik is supported by the German Research Foundation, project TORE (NI 369/18).

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk, Russian Federation

    René van Bevern & Oxana Yu. Tsidulko

  2. Algorithmics and Computational Complexity, Fakultät IV, TU Berlin, Berlin, Germany

    Till Fluschnik

  3. Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation

    Oxana Yu. Tsidulko

Authors
  1. René van Bevern
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  2. Till Fluschnik
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Correspondence to René van Bevern .

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Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  3. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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van Bevern, R., Fluschnik, T., Tsidulko, O.Y. (2019). On \((1+\varepsilon )\)-approximate Data Reduction for the Rural Postman Problem. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_20

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  • DOI: https://doi.org/10.1007/978-3-030-22629-9_20

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-22629-9

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