, Volume 71, Issue 1, pp 36–52 | Cite as

Approximating Minimum Manhattan Networks in Higher Dimensions

  • Aparna Das
  • Emden R. Gansner
  • Michael Kaufmann
  • Stephen Kobourov
  • Joachim Spoerhase
  • Alexander Wolff


We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in \({\mathbb {R}}^{d}\), find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair’s Manhattan (that is, L1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless \({\mathcal{P}} = \mathcal{NP}\)). Approximation algorithms are known for 2D, but not for 3D.

We present, for any fixed dimension d and any ε>0, an O(nε)-approximation algorithm. For 3D, we also give a 4(k−1)-approximation algorithm for the case that the terminals are contained in the union of k≥2 parallel planes.


Approximation algorithms Computational geometry Minimum Manhattan network 



This work was started at the 2009 Bertinoro Workshop on Graph Drawing. We thank the organizers Beppe Liotta and Walter Didimo for creating an inspiring atmosphere. We also thank Steve Wismath, Henk Meijer, Jan Kratochvíl, and Pankaj Agarwal for discussions. We are indebted to Stefan Felsner for pointing us to Soto and Telha’s work [18].


  1. 1.
    Arora, S.: Approximation schemes for NP-hard geometric optimization problems: a survey. Math. Program. 97(1–2), 43–69 (2003). doi:10.1007/s10107-003-0438-y MATHMathSciNetGoogle Scholar
  2. 2.
    Arya, S., Das, G., Mount, D.M., Salowe, J.S., Smid, M.: Euclidean spanners: short, thin, and lanky. In: Proc. 27th Annu. ACM Sympos. Theory Comput. (STOC’95), pp. 489–498 (1995). doi:10.1145/225058.225191 Google Scholar
  3. 3.
    Benkert, M., Wolff, A., Widmann, F., Shirabe, T.: The minimum Manhattan network problem: approximations and exact solutions. Comput. Geom. Theory Appl. 35(3), 188–208 (2006). doi:10.1016/j.comgeo.2005.09.004 CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Charikar, M., Chekuri, C., Cheung, T.Y., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation algorithms for directed Steiner problems. J. Algorithms 33(1), 73–91 (1999). doi:10.1006/jagm.1999.1042 CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Chepoi, V., Nouioua, K., Vaxès, Y.: A rounding algorithm for approximating minimum Manhattan networks. Theor. Comput. Sci. 390(1), 56–69 (2008). doi:10.1016/j.tcs.2007.10.013 CrossRefMATHGoogle Scholar
  6. 6.
    Chin, F., Guo, Z., Sun, H.: Minimum Manhattan network is NP-complete. Discrete Comput. Geom. 45, 701–722 (2011). doi:10.1007/s00454-011-9342-z CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Das, A., Gansner, E.R., Kaufmann, M., Kobourov, S., Spoerhase, J., Wolff, A.: Approximating minimum Manhattan networks in higher dimensions. ArXiv e-print arXiv:1107.0901v2 (2011)
  8. 8.
    Feldman, M., Kortsarz, G., Nutov, Z.: Improved approximating algorithms for directed Steiner forest. J. Comput. Syst. Sci. 78(1), 279–292 (2012). doi:10.1016/j.jcss.2011.05.009 CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Fuchs, B., Schulze, A.: A simple 3-approximation of minimum Manhattan networks. Tech. Rep. 570, Zentrum für Angewandte Informatik Köln (2008). See
  10. 10.
    Gudmundsson, J., Levcopoulos, C., Narasimhan, G.: Approximating a minimum Manhattan network. Nordic J. Comput. 8, 219–232 (2001). See MATHMathSciNetGoogle Scholar
  11. 11.
    Guo, Z., Sun, H., Zhu, H.: Greedy construction of 2-approximate minimum Manhattan networks. Int. J. Comput. Geom. Appl. 21(3), 331–350 (2011). doi:10.1142/S0218195911003688 CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Kato, R., Imai, K., Asano, T.: An improved algorithm for the minimum Manhattan network problem. In: Bose, P., Morin, P. (eds.) Proc. 13th Annu. Internat. Sympos. Algorithms Comput. (ISAAC’02). Lecture Notes Comput. Sci., vol. 2518, pp. 344–356. Springer, Berlin (2002). doi:10.1007/3-540-36136-7_31 Google Scholar
  13. 13.
    Lam, F., Alexandersson, M., Pachter, L.: Picking alignments from (Steiner) trees. J. Comput. Biol. 10, 509–520 (2003). doi:10.1089/10665270360688156 CrossRefGoogle Scholar
  14. 14.
    Lu, B., Ruan, L.: Polynomial time approximation scheme for the rectilinear Steiner arborescence problem. J. Comb. Optim. 4(3), 357–363 (2000). doi:10.1023/A:1009826311973 CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Muñoz, X., Seibert, S., Unger, W.: The minimal Manhattan network problem in three dimensions. In: Das, S., Uehara, R. (eds.) Proc. 3rd Internat. Workshop Algorithms Comput. (WALCOM’09). Lecture Notes Comput. Sci., vol. 5431, pp. 369–380. Springer, Berlin (2009). doi:10.1007/978-3-642-00202-1_32 Google Scholar
  16. 16.
    Nouioua, K.: Enveloppes de Pareto et réseaux de Manhattan: Caractérisations et algorithmes. Ph.D. thesis, Université de la Méditerranée (2005). See
  17. 17.
    Seibert, S., Unger, W.: A 1.5-approximation of the minimal Manhattan network problem. In: Deng, X., Du, D. (eds.) Proc. 16th Intern. Symp. Algorithms Comput. (ISAAC’05). Lecture Notes Comput. Sci., vol. 3827, pp. 246–255. Springer, Berlin (2005). doi:10.1007/11602613_26 Google Scholar
  18. 18.
    Soto, J.A., Telha, C.: Jump number of two-directional orthogonal ray graphs. In: Günlük, O., Woeginger, G. (eds.) Proc. 16th Internat. Conf. Integer Prog. Comb. Optimization (IPCO’11). Lecture Notes Comput. Sci., vol. 6655, pp. 389–403. Springer, Berlin (2011). doi:10.1007/978-3-642-20807-2_31 Google Scholar
  19. 19.
    Zelikovsky, A.: A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica 18(1), 99–110 (1997). doi:10.1007/BF02523690 CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Aparna Das
    • 1
  • Emden R. Gansner
    • 2
  • Michael Kaufmann
    • 3
  • Stephen Kobourov
    • 1
  • Joachim Spoerhase
    • 4
  • Alexander Wolff
    • 4
  1. 1.Dept. of Comp. Sci.University of ArizonaTucsonUSA
  2. 2.AT&T Labs ResearchFlorham ParkUSA
  3. 3.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany
  4. 4.Lehrstuhl I, Institut für InformatikUniversität WürzburgWürzburgGermany

Personalised recommendations