Journal of Combinatorial Optimization

, Volume 26, Issue 4, pp 709–722 | Cite as

Finding paths with minimum shared edges

  • Masoud T. Omran
  • Jörg-Rüdiger Sack
  • Hamid Zarrabi-Zadeh


Motivated by a security problem in geographic information systems, we study the following graph theoretical problem: given a graph G, two special nodes s and t in G, and a number k, find k paths from s to t in G so as to minimize the number of edges shared among the paths. This is a generalization of the well-known disjoint paths problem. While disjoint paths can be computed efficiently, we show that finding paths with minimum shared edges is NP-hard. Moreover, we show that it is even hard to approximate the minimum number of shared edges within a factor of \(2^{\log^{1-\varepsilon}n}\), for any constant ε>0. On the positive side, we show that there exists a (k−1)-approximation algorithm for the problem, using an adaption of a network flow algorithm. We design some heuristics to improve the quality of the output, and provide empirical results.


Minimum shared edges problem Approximation algorithm Inapproximability Heuristic algorithms 



The authors would like to thank Anil Maheshwari and Peter Widmayer for helpful discussions.


  1. Ahuja RK, Goldberg AV, Orlin JB, Tarjan RE (1992) Finding minimum-cost flows by double scaling. Math Program 53(1):243–266 MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bader D, Madduri K (2005) Design and implementation of the HPCS graph analysis benchmark on symmetric multiprocessors. In: Proc 12th internat conf high perform comput. Lecture notes comput sci, vol 3769, pp 465–476 Google Scholar
  3. Castanon DA (1990) Efficient algorithms for finding the k best paths through a trellis. IEEE Trans Aerosp Electron Syst 26(2):405–410 CrossRefGoogle Scholar
  4. Charikar M, Hajiaghayi M, Karloff H (2009) Improved approximation algorithms for label cover problems. In: Proc 17th annu European sympos algorithms. Lecture notes comput sci, vol 5757, pp 23–34 Google Scholar
  5. Even G, Kortsarz G, Slany W (2005) On network design problems: fixed cost flows and the covering steiner problem. ACM Trans Algorithms 1(1):74–101 MathSciNetCrossRefGoogle Scholar
  6. Garey M, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, New York zbMATHGoogle Scholar
  7. Goldberg AV, Rao S (1998) Beyond the flow decomposition barrier. J ACM 45(5):783–797 MathSciNetCrossRefzbMATHGoogle Scholar
  8. Itai A, Perl Y, Shiloach Y (1982) The complexity of finding maximum disjoint paths with length constraints. Networks 12(3):277–286 MathSciNetCrossRefzbMATHGoogle Scholar
  9. Kobayashi Y, Sommer C (2010) On shortest disjoint paths in planar graphs. Discrete Optim 7(4):234–245 MathSciNetCrossRefzbMATHGoogle Scholar
  10. Krumke SO, Noltemeier H, Schwarz S, Wirth H-C, Ravi R (1998) Flow improvement and network flows with fixed costs. In: Proc internat conf oper res: OR-98, pp 158–167 Google Scholar
  11. Lee S-W, Wu C-S (1999) A k-best paths algorithm for highly reliable communication networks. IEICE Trans Commun E82-B(4):586–590 Google Scholar
  12. Li C, McCormick TS, Simich-Levi D (1989) The complexity of finding two disjoint paths with min-max objective function. Discrete Appl Math 26(1):105–115 CrossRefGoogle Scholar
  13. Li C, McCormick ST, Simchi-Levi D (1992) Finding disjoint paths with different path-costs: complexity and algorithms. Networks 22(7):653–667 MathSciNetCrossRefzbMATHGoogle Scholar
  14. Nikolopoulos SD, Pitsillides A, Tipper D (1997) Addressing network survivability issues by finding the k-best paths through a trellis graph. In: Proc 16th IEEE internat conf comput commun, pp 370–377 Google Scholar
  15. Suurballe JW, Tarjan RE (1984) A quick method for finding shortest pairs of disjoint paths. Networks 14(2):325–336 MathSciNetCrossRefzbMATHGoogle Scholar
  16. Xu D, Chen Y, Xiong Y, Qiao C, He X (2006) On the complexity of and algorithms for finding the shortest path with a disjoint counterpart. IEEE/ACM Trans Netw 14(1):147–158 CrossRefGoogle Scholar
  17. Zheng SQ, Yang B, Yang M, Wang J (2007) Finding minimum-cost paths with minimum sharability. In: Proc 26th IEEE internat conf comput commun, pp 1532–1540 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Masoud T. Omran
    • 1
  • Jörg-Rüdiger Sack
    • 1
  • Hamid Zarrabi-Zadeh
    • 2
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Computer EngineeringSharif University of TechnologyTehranIran

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