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International Workshop on Graph-Theoretic Concepts in Computer Science

WG 2014: Graph-Theoretic Concepts in Computer Science pp 250–262Cite as

Parameterized Directed \(k\)-Chinese Postman Problem and \(k\) Arc-Disjoint Cycles Problem on Euler Digraphs

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 8747))

Abstract

In the Directed \(k\)-Chinese Postman Problem (\(k\)-DCPP), we are given a connected weighted digraph \(G\) and asked to find \(k\) non-empty closed directed walks covering all arcs of \(G\) such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513, 124–128 (2013)) asked for the parameterized complexity of \(k\)-DCPP when \(k\) is the parameter. We prove that the \(k\)-DCPP is fixed-parameter tractable.

We also consider a related problem of finding \(k\) arc-disjoint directed cycles in an Euler digraph, parameterized by \(k\). Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs.

A full version of this paper appears at http://arxiv.org/abs/1402.2137.

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Notes

  1. 1.

    For terminology and results on parameterized algorithms and complexity we refer the reader to, e.g., the monograph [4].

  2. 2.

    Here, as in the proof, the union of multigraphs means that the multiplicity of an arc in the union equals the sum of multiplicities of this arc in the multigraphs of the union.

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Acknowledgement

Research of GG was supported by Royal Society Wolfson Research Merit Award.

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Authors and Affiliations

  1. Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK

    Gregory Gutin, Mark Jones, Bin Sheng & Magnus Wahlström

Authors
  1. Gregory Gutin
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  2. Mark Jones
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  3. Bin Sheng
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  4. Magnus Wahlström
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Corresponding author

Correspondence to Gregory Gutin .

Editor information

Editors and Affiliations

  1. LITA, Université de Lorraine, Metz Cedex 01, France

    Dieter Kratsch

  2. Université d'Orléans, Orléans, France

    Ioan Todinca

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Gutin, G., Jones, M., Sheng, B., Wahlström, M. (2014). Parameterized Directed \(k\)-Chinese Postman Problem and \(k\) Arc-Disjoint Cycles Problem on Euler Digraphs. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_21

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  • DOI: https://doi.org/10.1007/978-3-319-12340-0_21

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12339-4

  • Online ISBN: 978-3-319-12340-0

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