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)


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.


Eulerian extension Lossy kernelization Parameterized complexity 



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).


  1. 1.
    Belenguer, J.M., Benavent, E., Lacomme, P., Prins, C.: Lower and upper bounds for the mixed capacitated arc routing problem. Comput. Oper. Res. 33(12), 3363–3383 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    van Bevern, R., Fluschnik, T., Tsidulko, O.Yu.: Parameterized algorithms and data reduction for safe convoy routing. In: Proceedings of 18th ATMOS, OASIcs, vol. 65, pp. 10:1–10:19. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2018)Google Scholar
  3. 3.
    van Bevern, R., Hartung, S., Nichterlein, A., Sorge, M.: Constant-factor approximations for capacitated arc routing without triangle inequality. Oper. Res. Lett. 42(4), 290–292 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    van Bevern, R., Komusiewicz, C., Sorge, M.: A parameterized approximation algorithm for the mixed and windy capacitated arc routing problem: theory and experiments. Networks 70(3), 262–278 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    van Bevern, R., Niedermeier, R., Sorge, M., Weller, M.: Complexity of arc routing problems. In: Arc Routing: Problems, Methods, and Applications, MOS-SIAM Series on Optimization, vol. 20. SIAM (2014)Google Scholar
  6. 6.
    Brandão, J., Eglese, R.: A deterministic tabu search algorithm for the capacitated arc routing problem. Comput. Oper. Res. 35(4), 1112–1126 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Corberán, Á., Laporte, G. (eds.): Arc Routing: Problems, Methods, and Applications. SIAM, Philadelphia (2014)zbMATHGoogle Scholar
  8. 8.
    Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). Scholar
  9. 9.
    Dorn, F., Moser, H., Niedermeier, R., Weller, M.: Efficient algorithms for Eulerian extension and Rural Postman. SIAM J. Discrete Math. 27(1), 75–94 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Edmonds, J., Johnson, E.L.: Matching, Euler tours and the Chinese postman. Math. Program. 5, 88–124 (1973)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Eiben, E., Hermelin, D., Ramanujan, M.S.: Lossy kernels for hitting subgraphs. In: Proceedings of 42nd MFCS, LIPIcs, vol. 83, pp. 67:1–67:14. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl (2017)Google Scholar
  12. 12.
    Eiben, E., Kumar, M., Mouawad, A.E., Panolan, F., Siebertz, S.: Lossy kernels for connected dominating set on sparse graphs. In: Proceedings of 35th STACS, LIPIcs, vol. 96, pp. 29:1–29:15. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2018)Google Scholar
  13. 13.
    Eiselt, H.A., Gendreau, M., Laporte, G.: Arc routing problems, Part II: The Rural Postman Problem. Oper. Res. 43(3), 399–414 (1995)CrossRefGoogle Scholar
  14. 14.
    Etscheid, M., Kratsch, S., Mnich, M., Röglin, H.: Polynomial kernels for weighted problems. J. Comput. Syst. Sci. 84, 1–10 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fellows, M.R., Kulik, A., Rosamond, F.A., Shachnai, H.: Parameterized approximation via fidelity preserving transformations. J. Comput. Syst. Sci. 93, 30–40 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ghiani, G., Improta, G.: The laser-plotter beam routing problem. J. Oper. Res. Soc. 52(8), 945–951 (2001)CrossRefGoogle Scholar
  18. 18.
    Ghiani, G., Laporte, G.: Eulerian location problems. Networks 34(4), 291–302 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Golden, B.L., Wong, R.T.: Capacitated arc routing problems. Networks 11(3), 305–315 (1981)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Grötschel, M., Jünger, M., Reinelt, G.: Optimal control of plotting and drilling machines: a case study. Z. Oper. Res. 35(1), 61–84 (1991)zbMATHGoogle Scholar
  21. 21.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  22. 22.
    Gutin, G., Wahlström, M., Yeo, A.: Rural Postman parameterized by the number of components of required edges. J. Comput. Syst. Sci. 83(1), 121–131 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hermelin, D., Kratsch, S., Sołtys, K., Wahlström, M., Wu, X.: A completeness theory for polynomial (Turing) kernelization. Algorithmica 71(3), 702–730 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jansen, K.: Bounds for the general capacitated routing problem. Networks 23(3), 165–173 (1993)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Karpinski, M., Lampis, M., Schmied, R.: New inapproximability bounds for TSP. J. Comput. Syst. Sci. 81(8), 1665–1677 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kratsch, S.: Recent developments in kernelization: A survey. Bull. EATCS 113 (2014)Google Scholar
  27. 27.
    Krithika, R., Majumdar, D., Raman, V.: Revisiting connected vertex cover: FPT algorithms and lossy kernels. Theor. Comput. Syst. 62(8), 1690–1714 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Krithika, R., Misra, P., Rai, A., Tale, P.: Lossy kernels for graph contraction problems. In: Proceedings 36th FSTTCS, LIPIcs, vol. 65, pp. 23:1–23:14. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl (2016)Google Scholar
  29. 29.
    Lokshtanov, D., Panolan, F., Ramanujan, M.S., Saurabh, S.: Lossy kernelization. In: Proceedings 49th STOC, pp. 224–237. ACM (2017)Google Scholar
  30. 30.
    Marx, D., Végh, L.A.: Fixed-parameter algorithms for minimum-cost edge-connectivity augmentation. ACM Trans. Algorithms 11(4), 27:1–27:24 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Orloff, C.S.: A fundamental problem in vehicle routing. Networks 4(1), 35–64 (1974)MathSciNetCrossRefGoogle Scholar
  32. 32.
    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). Scholar
  33. 33.
    Sorge, M., van Bevern, R., Niedermeier, R., Weller, M.: A new view on Rural Postman based on Eulerian extension and matching. J. Discrete Algorithms 16, 12–33 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ulusoy, G.: The fleet size and mix problem for capacitated arc routing. Eur. J. Oper. Res. 22(3), 329–337 (1985)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wøhlk, S.: An approximation algorithm for the capacitated arc routing problem. Open Oper. Res. J. 2, 8–12 (2008)MathSciNetCrossRefGoogle Scholar

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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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