On Residual Approximation in Solution Extension Problems

  • Mathias Weller
  • Annie Chateau
  • Rodolphe Giroudeau
  • Jean-Claude König
  • Valentin Pollet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10043)


The solution extension variant of a problem consists in, being given an instance and a partial solution, finding the best solution comprising the given partial solution. Many problems have been studied with a similar approach. For instance the Precoloring Extension problem, the clustered variant of the Travelling Salesman problem, or the General Routing Problem are in a way typical examples of solution extension variant problems. Motivated by practical applications of such variants, this work aims to explore different aspects around extension on classical optimization problems. We define residue-approximations as algorithms whose performance ratio on the non-prescribed part can be bounded, and corresponding complexity classes. Using residue-approximation, we classify several problems according to their residue-approximability.


  1. 1.
    Anily, S., Bramel, J., Hertz, A.: A 5/3-approximation algorithm for the clustered traveling salesman tour and path problems. Oper. Res. Lett. 24(1–2), 29–35 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arkin, E.M., Hassin, R., Klein, L.: Restricted delivery problems on a network. Networks 29(4), 205–216 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avidor, A., Zwick, U.: Approximating MIN 2-SAT and MIN 3-SAT. Theory Comput. Syst. 38(3), 329–345 (2005). ISSN 1433–0490MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discrete Math. 12(3), 289–297 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bar-Yehuda, R., Even, S.: A linear-time approximation algorithm for the weighted vertex cover problem. J. Algorithms 2(2), 198–203 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Biró, M., Hujter, M., Tuza, Z.: Precoloring extension. i. interval graphs. Discrete Math. 100(1–3), 267–279 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Björklund, A., Husfeldt, T., Taslaman, N.: Shortest cycle through specified elements. In Proceedings of 23rd SODA, pp. 1747–1753 (2012)Google Scholar
  8. 8.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. TR 388, Graduate School of Industrial Administration. Carnegie Mellon University (1976)Google Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  10. 10.
    Gendreau, M., Laporte, G., Hertz, A.: An approximation algorithm for the traveling salesman problem with backhauls. Oper. Res. 45(4), 639–641 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goemans, M.X., Williamson, D.P.: New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM J. Discrete Math. 7(4), 656–666 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gusfield, D., Pitt, L.: A bounded approximation for the minimum cost 2-SAT problem. Algorithmica 8(2), 103–117 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Guttmann-Beck, N., Hassin, R., Khuller, S., Raghavachari, B.: Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem. Algorithmica 28(4), 422–437 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hujter, M., Tuza, Z.: Precoloring extension. ii. graphs classes related to bipartite graphs. Acta Math. Univ. Comenian. (N.S.) 62(1), 1–11 (1993)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Jansen, K.: An approximation algorithm for the general routing problem. Inf. Process. Lett. 41(6), 333–339 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Knauer, M., Spoerhase, J.: Better approximation algorithms for the maximum internal spanning tree problem. Algorithmica 71(4), 797–811 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marx, D.: Precoloring extension on unit interval graphs. Discrete Appl. Math. 154(6), 995–1002 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Orloff, C.S.: A fundamental problem in vehicle routing. Networks 4(1), 35–64 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Robins, G., Zelikovsky, A.: Improved steiner tree approximation in graphs. In: Proceedings of 11th SODA, pp. 770–779 (2000)Google Scholar
  20. 20.
    Simchi-Levi, D.: New worst-case results for the bin packing problem. Naval Res. Logist. 41, 579–585 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Weller, M., Chateau, A., Giroudeau, R.: Exact approaches for scaffolding. BMC Bioinform. 16(Suppl. 14), S2 (2015)CrossRefGoogle Scholar
  22. 22.
    Weller, M., Chateau, A., Giroudeau, R.: On the complexity of scaffolding problems: from cliques to sparse graphs. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 409–423. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-26626-8_30 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Mathias Weller
    • 1
    • 2
  • Annie Chateau
    • 1
    • 2
  • Rodolphe Giroudeau
    • 1
  • Jean-Claude König
    • 1
  • Valentin Pollet
    • 1
  1. 1.LIRMM - CNRS UMR 5506MontpellierFrance
  2. 2.IBCMontpellierFrance

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