Advertisement

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

  • Gregory GutinEmail author
  • Mark Jones
  • Bin Sheng
  • Magnus Wahlström
Conference paper
Part of the Lecture Notes in Computer Science book series (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.

Notes

Acknowledgement

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

References

  1. 1.
    Bang-Jensen, J., Gutin, G.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer, New York (2009)CrossRefGoogle Scholar
  2. 2.
    van Bevern, R., Niedermeier, R., Sorge, M., Weller, M.: Complexity of arc routing problems. In: Corberán, A., Laporte, G. (eds.) Arc Routing: Problems, Methods and Applications, SIAM, Phil. (in press)Google Scholar
  3. 3.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5), 1–19 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, London (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dorn, F., Moser, H., Niedermeier, R., Weller, M.: Efficient algorithms for Eulerian extension. SIAM J. Discrete Math. 27(1), 75–94 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Edmonds, J., Johnson, E.L.: Matching, Euler tours and the Chinese postman. Math. Program. 5, 88–124 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grohe, M., Grüber, M.: Parameterized approximability of the disjoint cycle problem. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 363–374. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Gutin, G., Muciaccia, G., Yeo, A.: Parameterized complexity of \(k\)-Chinese postman problem. Theor. Comput. Sci. 513, 124–128 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lin, Y., Zhao, Y.: A new algorithm for the directed Chinese postman problem. Comput. Oper. Res. 15(6), 577–584 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lucchesi, C.L.: A minimax equality for directed graphs. Ph.D. thesis, University of Waterloo, Ontario, Canada (1976)Google Scholar
  11. 11.
    Pearn, W.L.: Solvable cases of the \(k\)-person Chinese postman problem. Oper. Res. Lett. 16(4), 241–244 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Reed, B., Robertson, N., Seymour, P.D., Thomas, R.: Packing directed circuits. Combinatorica 16(4), 535–554 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Seymour, P.D.: Packing circuits in Eulerian digraphs. Combinatorica 16(2), 223–231 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Slivkins, A.: Parameterized tractability of edge-disjoint paths on directed acyclic graphs. SIAM J. Discrete Math. 24(1), 146–157 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sorge, M.: Some Algorithmic Challenges in Arc Routing. Talk at NII Shonan Seminar no. 18, May 2013Google Scholar
  16. 16.
    Sorge, M., van Bevern, R., Niedermeier, R., Weller, M.: From few components to an Eulerian graph by adding arcs. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 307–318. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Sorge, M., van Bevern, R., Niedermeier, R., Weller, M.: A new view on Rural Postman based on Eulerian Extension and Matching. J. Discrete Alg. 16, 12–33 (2012)CrossRefzbMATHGoogle Scholar
  18. 18.
    Thomassen, C.: On the complexity of finding a minimum cycle cover of a graph. SIAM J. Comput. 26(3), 675–677 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Vygen, J.: NP-completeness of some edge-disjoint paths problems. Discrete Appl. Math. 61(1), 83–90 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhang, L.: Polynomial algorithms for the \(k\)-Chinese postman problem. In: Information Processing ’92, vol. 1, pp. 430–435 (1992)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Gregory Gutin
    • 1
    Email author
  • Mark Jones
    • 1
  • Bin Sheng
    • 1
  • Magnus Wahlström
    • 1
  1. 1.Royal Holloway, University of LondonEgham, SurreyUK

Personalised recommendations