Approximation of the Degree-Constrained Minimum Spanning Hierarchies

  • Miklós Molnár
  • Sylvain Durand
  • Massinissa Merabet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8576)


Degree-constrained spanning problems are well known and are mainly used to solve capacity constrained routing problems. The degree-constrained spanning tree problems are NP-hard and computing the minimum cost spanning tree is not approximable. Often, applications (such as some degree-constrained communications) do not need trees as solutions. Recently, a more flexible, connected, graph related structure called hierarchy was proposed to span a set of vertices under constraints. This structure permits a new formulation of some degree-constrained spanning problems. In this paper we show that although the newly formulated problem is still NP-hard, it is approximable with a constant ratio. In the worst case, this ratio is bounded by 3/2. We provide a simple heuristic and prove its approximation ratio is the best possible for any algorithm based on a minimum spanning tree.


Graph theory Networks Degree-Constrained Spanning Problem Spanning Hierarchy Approximation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Papadimitriou, C.H., Yannakakis, M.: The Complexity of Restricted Minimum Spanning Tree Problems (Extended Abstract). In: Maurer, H.A. (ed.) ICALP 1979. LNCS, vol. 71, pp. 460–470. Springer, Heidelberg (1979)Google Scholar
  2. 2.
    Cieslik, D.: The vertex degrees of minimum spanning trees. European Journal of Operational Research 125, 278–282 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ruzika, S., Hamacher, H.W.: A Survey on Multiple Objective Minimum Spanning Tree Problems. In: Lerner, J., Wagner, D., Zweig, K.A. (eds.) Algorithmics. LNCS, vol. 5515, pp. 104–116. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Hunt III, H.B.: Approximation algorithms for degree-constrained minimum-cost network-design problems. Algorithmica 31, 58–78 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Deo, N., Hakimi, S.: The shortest generalized Hamiltonian tree. In: Sixth Annual Allerton Conference, pp. 879–888 (1968)Google Scholar
  6. 6.
    Molnár, M.: Hierarchies to Solve Constrained Connected Spanning Problems. Technical Report 11029, LIRMM (2011)Google Scholar
  7. 7.
    Merabet, M., Durand, S., Molnar, M.: Exact solution for bounded degree connected spanning problems. Technical Report 12027, Laboratoire d’Informatique de Robotique et de Microélectronique de Montpellier - LIRMM (2012)Google Scholar
  8. 8.
    Zhou, Y., Poo, G.S.: Optical multicast over wavelength-routed wdm networks: A survey. Optical Switching and Networking 2, 176–197 (2005)CrossRefGoogle Scholar
  9. 9.
    Ali, M., Deogun, J.: Cost-effective implementation of multicasting in wavelength-routed networks. IEEE J. Lightwave Technol., Special Issue on Optical Networks 18, 1628–1638 (2000)CrossRefGoogle Scholar
  10. 10.
    Obruca, A.K.: Spanning tree manipulation and the travelling salesman problem. The Computer Journal 10, 374–377 (1968)CrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  12. 12.
    Ravi, R., Marathe, M.V., Ravi, S.S., Rosenkrantz, D.J., Hunt III, H.B.: Many birds with one stone: multi-objective approximation algorithms. In: Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing, STOC 1993, pp. 438–447. ACM, New York (1993)CrossRefGoogle Scholar
  13. 13.
    Fürer, M., Raghavachari, B.: Approximating the minimum degree spanning tree to within one from the optimal degree. In: Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1992, pp. 317–324. Society for Industrial and Applied Mathematics, Philadelphia (1992)Google Scholar
  14. 14.
    Goemans, M.: Minimum bounded degree spanning trees. In: 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006, pp. 273–282 (2006)Google Scholar
  15. 15.
    Singh, M., Lau, L.C.: Approximating minimum bounded degree spanning trees to within one of optimal. In: STOC 2007: Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, pp. 661–670. ACM, New York (2007)Google Scholar
  16. 16.
    Hoogeveen, J.A.: Analysis of Christofides’ heuristic: Some paths are more difficult than cycles. Oper. Res. Lett. 10, 291–295 (1991)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Miklós Molnár
    • 1
  • Sylvain Durand
    • 1
  • Massinissa Merabet
    • 1
  1. 1.Laboratory LIRMM UMR 5506, CC477University Montpellier 2Montpellier Cedex 5France

Personalised recommendations