Journal of Combinatorial Optimization

, Volume 19, Issue 2, pp 200–216 | Cite as

Complexity analysis for maximum flow problems with arc reversals

  • Steffen Rebennack
  • Ashwin Arulselvan
  • Lily Elefteriadou
  • Panos M. PardalosEmail author


We provide a comprehensive study on network flow problems with arc reversal capabilities. The problem is to identify the arcs to be reversed in order to achieve a maximum flow from source(s) to sink(s). The problem finds its applications in emergency transportation management, where the lanes of a road network could be reversed to enable flow in the opposite direction. We study several network flow problems with the arc reversal capability and discuss their complexity. More specifically, we discuss the polynomial time algorithms for the maximum dynamic flow problem with arc reversal capability having a single source and a single sink, and for the maximum (static) flow problem. The presented algorithms are based on graph transformations and reductions to polynomially solvable flow problems. In addition, we show that the quickest transshipment problem with arc reversal capability and the problem of minimizing the total cost resulting from arc switching costs are \(\mathcal{NP}\) -hard.


Contraflow problem Dynamic contraflow problem Complexity analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ahuja RK, Magnati TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice Hall, Englewood Cliffs Google Scholar
  2. Bern MW, Lawler EL, Wong AL (1987) Linear-time computation of optimal subgraph of decomposable graphs. J Algorithms 8:216–235 zbMATHCrossRefMathSciNetGoogle Scholar
  3. Burkard R, Dlaska K, Klinz B (1993) The quickest flow problem. Math Methods Oper Res 37(1):31–58 zbMATHCrossRefMathSciNetGoogle Scholar
  4. Cheriyan J, Maheshwari SN (1989) Analysis of preflow push algorithm for maximum network flow. SIAM J Comput 18:1057–1086 zbMATHCrossRefMathSciNetGoogle Scholar
  5. Ford FR, Fulkerson DR (1962) Flows in networks. Princeton University Press, Princeton zbMATHGoogle Scholar
  6. Garey MR, Johnson DS (1979) Computers and intractability—a guide to the theory of NP-completeness. Freeman, New York zbMATHGoogle Scholar
  7. Goldberg AV, Tarjan RE (1989) Finding minimum-cost circulations by cancelling negative cycles. J ACM 36:873–886 zbMATHCrossRefMathSciNetGoogle Scholar
  8. Gross JL, Yellen J (2003) Handbook of graph theory. Discrete mathematics and its applications. CRC, New York Google Scholar
  9. Guisewite GM, Pardalos PM (1990) Minimum concave-cost network flow problems: applications, complexity, and algorithms. Ann Oper Res 25:75–100 zbMATHCrossRefMathSciNetGoogle Scholar
  10. Hajek B, Ogier RG (1984) Optimal dynamic routing in communication networks with continuous traffic. Networks 14:457–487 zbMATHCrossRefMathSciNetGoogle Scholar
  11. Hamza-Lup GL, Hua K, Lee M, Peng R (2004) Enhancing intelligent transportation systems to improve and support homeland security. In: Proceedings of the 7th international IEEE conference on intelligent transportation systems, pp 250–255 Google Scholar
  12. Hoppe BE (1995) Efficient dynamic network flow algorithms. PhD thesis, Cornell University.
  13. Kim D, Pardalos PM (2000) Dynamic slope scaling and trust interval techniques for solving concave piecewise linear network flow problems. Networks 35(3):216–222 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Kim S, Shekhar S (2005) Contraflow network reconfiguration for evaluation planning: a summary of results. In: Proceedings of the 13th annual ACM international workshop on geographic information systems, pp 250–259 Google Scholar
  15. Krumke S, Noltemeier H, Schwarz S, Wirth H, Ravi R (1998) Flow improvement and network flows with, fixed costs. In: Proceedings of the international conference of operations research (0R’98), Zürich, pp 158–167 Google Scholar
  16. Megiddo N (1979) Combinatorial optimization with rational objective functions. Math Oper Res 4:414–424 zbMATHCrossRefMathSciNetGoogle Scholar
  17. Melkonian V (2007) Flows in dynamic networks with aggregate arc capacities. Inf Process Lett 101(1):30–35 zbMATHCrossRefMathSciNetGoogle Scholar
  18. Orlin JB (1983) Maximum-throughput dynamic network flows. Math Program 27:214–231 zbMATHCrossRefMathSciNetGoogle Scholar
  19. Theodoulou G, Wolshon B (2004) Alternative methods to increase the effectiveness of freeway contraflow evacuation. J Transp Res Board 1865:48–56 CrossRefGoogle Scholar
  20. Tuydes H, Ziliaskopoulos A (2006) Tabu-based heuristic approach for optimization of network evacuation contraflow. Transp Res Record 1964:157–168 CrossRefGoogle Scholar
  21. Williams B, Tagliaferri A, Meinhold S, Hummer J, Rouphail N (2007) Simulation and analysis of freeway lane reversal for coastal hurricane evacuation. J Urban Plng Devel 133(1):61–72 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Steffen Rebennack
    • 1
  • Ashwin Arulselvan
    • 1
  • Lily Elefteriadou
    • 2
  • Panos M. Pardalos
    • 1
    Email author
  1. 1.Department of Industrial & Systems Engineering, Center for Applied OptimizationUniversity of FloridaGainesvilleUSA
  2. 2.Department of Civil & Coastal Engineering, Transportation Research CenterUniversity of FloridaGainesvilleUSA

Personalised recommendations