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Annals of Operations Research

, Volume 223, Issue 1, pp 379–401 | Cite as

Location of speed-up subnetworks

  • Marie SchmidtEmail author
  • Anita Schöbel
Article

Abstract

Let a network with edge weights, a set of point-to-point transportation requests and a factor \(\alpha \) be given. Our goal is to design a subnetwork of given length along which transportation costs are reduced by \(\alpha \). This reduces the costs of the network traffic which will choose to use edges of the new subnetwork if this is the more efficient option. Our goal is to design the subnetwork in such a way that the worst-case cost of all routing requests is minimized. The problem occurs in many applications, among others in transportation networks, in backbone, information, communication, or electricity networks. We classify the problem according to the types of the given network and of the network to be established. We are able to clarify the complexity status in all considered cases. It turns out that finding an optimal subtree in a tree already is NP-hard. We therefore further investigate this case and propose results and a solution approach.

References

  1. Chung, F. (1986). Diameters of communication networks. In Mathematics of information processing. Proceedings of Symposia in Applied Mathematics, 34, pp. 1–18.Google Scholar
  2. Contreras, I., & Fernández, P. (2012). General network design: A unified view of combined location and network design problems. European Journal of Operational Research, 219(3), 680–697.CrossRefGoogle Scholar
  3. Demaine, E. & Zadimoghaddam, M. (2010). Minmizing the diameter of a network using shortcut edges. In: Algorithm theory: SWAT 2010, Lecture Notes in Computer Science (Vol. 6139, pp. 420–431), New York: Springer.Google Scholar
  4. Eggemann, N., & Noble, S. (2009). Minimizing the oriented diameter of a planar graph. Electronic Notes in Discrete Mathematics, 34, 267–271.CrossRefGoogle Scholar
  5. Engelhardt-Funke, O., & Kolonko, M. (2001). Cost-benefit analysis of investments into railway networks with randomly pertubed operations. In S. Voß & J. Daduna (Eds.), Computer-aided transit scheduling, Lecture Notes in Economics and Mathematical Systems (Vol. 505, pp. 442–459). New York: Springer.Google Scholar
  6. Fernández, P., & Marín, A. (2003). A heuristic procedure for path location with multisource demand. Information Systems and Operational Research, 41(2), 165–177.Google Scholar
  7. Garey, M., & Johnson, D. (1979). Computers and intractability—A guide to the theory of NP-completeness. San Francisco: Freeman.Google Scholar
  8. Gutierrez, G., Donoso, M., Obreque, C., & Marianov, V. (2010). Minimum cost path location for maximum traffic capture. Computers & Industrial Engineering, 58(2), 332–341.CrossRefGoogle Scholar
  9. Hakimi, S. L., Schmeichel, E. F., & Labbé, M. (1993). On locating path- or tree-shaped facilities on networks. Networks, 23(6), 543–555.CrossRefGoogle Scholar
  10. Hedetniemi, S. M., Cockayne, E. J., & Hedetniemi, S. T. (1981). Linear algorithms for finding the jordan center and path center of a tree. Transportation Science, 15(2), 98–114.Google Scholar
  11. Hofmann, H. (2009). Heuristiken zur Platzierung von Bäumen in Bäumen (2009). Bachelors Thesis, (in German).Google Scholar
  12. Koster, A. & Munoz, X. (Eds.). (2010). Graphs and algorithms in communication networks—studies in broadband, optical, wireless and Ad Hoc networks. Texts in Theoretical Computer Science. An EATCS Series. New York: Springer.Google Scholar
  13. Labbé, M., & Laporte, G. (2005). Locating median cycles in networks. European Journal of Operational Research, 160(2), 457–470.CrossRefGoogle Scholar
  14. Lamb, J. D. (2010a). Insertion heuristic for central cycle problems. Networks, 56(1), 70–80.Google Scholar
  15. Lamb, J. D. (2010b). Variable neighbourhood structures for cycle location problems. European Journal of Operational Research, 223(1), 15–26.Google Scholar
  16. Laporte, G., Marín, A., Mesa, J., & Ortega, F. (2006). An integrated methodology for rapid transit network design. Algorithmic methods for railway optimization, Lecture Notes on Computer Science. New York: Springer.Google Scholar
  17. Laporte, G., Mesa, J., Ortega, F., & Perea, F. (2011). Planning rapid transit networks. Socio-Economic Planning Sciences, 45(3), 95–104.CrossRefGoogle Scholar
  18. Mesa, J. A., & Boffey, T. (1996). A review of extensive facility location in networks. European Journal of Operational Research, 95(3), 592–603.CrossRefGoogle Scholar
  19. Meyerson, A., & Tagiku, B. (2009). Minimizing the average shortest path distances via shortcur edge addition. Proceedings of the International Workshop on Approximation Algorithms for Combinatorial optimization Problems, Lecture Notes in Computer Science, 5687, 272–285.Google Scholar
  20. Nickel, S., Schöbel, A., & Sonneborn, T. (2001). Hub location problems in urban traffic networks. In Pursula Niittymäki (Ed.), Mathematical methods and optimization in transportation systems (pp. 95–107). Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
  21. Puerto, J., Ricca, F., & Scozzari, A. (2012). Range minimization problems in path-facility location on trees. Discrete Applied Mathematics, 160(15), 2294–2305.CrossRefGoogle Scholar
  22. Puerto, J., & Tamir, A. (2007). Locating tree-shaped facilities using the ordered median objective. Mathematical Programming, Series A, 102(2), 313–338.CrossRefGoogle Scholar
  23. Ruzika, S. & Thiemann, M. (2011). Reliable and restricted quickest path problems. In J. Pahl, T. Reiners & S. Vo (Eds.) INOC 2011, no. 6701 in LNCS, pp. 309–314. Heidelberg: Springer.Google Scholar
  24. Schmidt, M. (2009). Netzwerkstandortprobleme mit OD-Paaren. Master’s thesis, Georg August Universität Göttingen, Göttingen (in German).Google Scholar
  25. Schöbel, A. (2012). Line planning in public transportation: Models and methods. OR Spectrum, 34(3), 491–510.Google Scholar
  26. Székely, L. A., & Wang, H. (2005). On subtrees of trees. Advances in Applied Mathematics, 34(1), 138–155.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute for Numerical and Applied MathematicsGeorg-August University GöttingenGöttingenGermany

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