Approximation Schemes for Capacitated Geometric Network Design

  • Anna Adamaszek
  • Artur Czumaj
  • Andrzej Lingas
  • Jakub Onufry Wojtaszczyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


We study a capacitated network design problem in geometric setting. We assume that the input consists of an integral link capacity k and two sets of points on a plane, sources and sinks, each source/sink having an associated integral demand (amount of flow to be shipped from/to). The capacitated geometric network design problem is to construct a minimum-length network N that allows to route the requested flow from sources to sinks, such that each link in N has capacity k; the flow is splittable and parallel links are allowed in N.

The capacitated geometric network design problem generalizes, among others, the geometric Steiner tree problem, and as such it is NP-hard.

We show that if the demands are polynomially bounded and the link capacity k is not too large, the single-sink capacitated geometric network design problem admits a polynomial-time approximation scheme. If the capacity is arbitrarily large, then we design a quasi-polynomial time approximation scheme for the capacitated geometric network design problem allowing for arbitrary number of sinks. Our results rely on a derivation of an upper bound on the number of vertices different from sources and sinks (the so called Steiner vertices) in an optimal network. The bound is polynomial in the total demand of the sources.


Approximation Scheme Network Design Total Demand Link Capacity Network Design Problem 
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  1. 1.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM 45(5), 753–782 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Czumaj, A., Czyzowicz, J., Gąsieniec, L., Jansson, J., Lingas, A., Zylinski, P.: Approximation algorithms for buy-at-bulk geometric network design. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 168–180. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Czumaj, A., Lingas, A.: On approximability of the minimum-cost k-connected spanning subgraph problem. In: Proc. 10th SODA, pp. 281–290 (1999)Google Scholar
  4. 4.
    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
  5. 5.
    Haimovich, M., Rinnooy Kan, A.H.G.: Bounds and heuristics for capacitated routing problems. Mathematics of Operation Research 10(4), 527–542 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hassin, R., Ravi, R., Salman, F.S.: Approximation algorithms for a capacitated network design problem. Algorithmica 38, 417–431 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. Annals of Discrete Mathematics, vol. 53. North-Holland, Amsterdam (1992)zbMATHGoogle Scholar
  8. 8.
    Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing 28(4), 1298–1309 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Morsy, E., Nagamochi, H.: Approximation to the minimum cost edge installation problem. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 292–303. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Salman, F.S., Cheriyan, J., Ravi, R., Subramanian, S.: Approximating the single-sink link-installation problem in network design. SIAM J. Optimization 11(3), 595–610 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Zachariasen, M.: A catalog of Hanan grid problems. Networks 38(2), 76–83 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Anna Adamaszek
    • 1
  • Artur Czumaj
    • 1
  • Andrzej Lingas
    • 2
  • Jakub Onufry Wojtaszczyk
    • 3
  1. 1.DIMAP and Department of Computer ScienceUniversity of WarwickUK
  2. 2.Department of Computer ScienceLund UniversitySweden
  3. 3.Google, Inc.USA

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