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

  • René van BevernEmail author
  • Till Fluschnik
  • Oxana Yu. Tsidulko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Eulerian extension Lossy kernelization Parameterized complexity 

Notes

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsNovosibirsk State UniversityNovosibirskRussian Federation
  2. 2.Algorithmics and Computational Complexity, Fakultät IVTU BerlinBerlinGermany
  3. 3.Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of SciencesNovosibirskRussian Federation

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