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Algorithms for the Simple Equal Flow Problem

Author

Listed:
  • Ravindra K. Ahuja

    (Department of Industrial & Systems Engineering, University of Florida, Gainesville, Florida 32611)

  • James B. Orlin

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Giovanni M. Sechi

    (Department of Hydraulics, University of Cagliari, 09123 Cagliari, Sardinia, Italy)

  • Paola Zuddas

    (Department of Hydraulics, University of Cagliari, 09123 Cagliari, Sardinia, Italy)

Abstract

The minimum cost flow problem is to determine a least cost shipment of a commodity through a network G = (N, A) in order to satisfy demands at certain nodes from available supplies at other nodes. In this paper, we study a variant of the minimum cost flow problem where we are given a set R \subseteq A of arcs and require that each arc in R must carry the same amount of flow. This problem, which we call the simple equal flow problem, arose while modeling a water resource system management in Sardinia, Italy. We consider the simple equal flow problem in a directed network with n nodes, m arcs, and where all arc capacities and node supplies are integer and bounded by U. We develop several algorithms for the simple equal flow problem---the network simplex algorithm, the parametric simplex algorithm, the combinatorial parametric algorithm, the binary search algorithm, and the capacity scaling algorithm. The binary search algorithm solves the simple equal flow problem in O(log(nU)) applications of any minimum cost flow algorithm. The capacity scaling algorithm solves it in O(m(m + n logn) log (nU)) time, which is almost the same time needed to solve the minimum cost flow problem by the capacity scaling algorithm. These algorithms can be easily modified to obtain an integer solution of the simple equal flow problem.

Suggested Citation

  • Ravindra K. Ahuja & James B. Orlin & Giovanni M. Sechi & Paola Zuddas, 1999. "Algorithms for the Simple Equal Flow Problem," Management Science, INFORMS, vol. 45(10), pages 1440-1455, October.
    • Handle: RePEc:inm:ormnsc:v:45:y:1999:i:10:p:1440-1455
    DOI: 10.1287/mnsc.45.10.1440
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    References listed on IDEAS

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    1. Éva Tardos, 1986. "A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs," Operations Research, INFORMS, vol. 34(2), pages 250-256, April.
    2. Ali, Agha Iqbal & Kennington, Jeff & Shetty, Bala, 1988. "The equal flow problem," European Journal of Operational Research, Elsevier, vol. 36(1), pages 107-115, July.
    3. Carraresi, P. & Gallo, G., 1984. "Network models for vehicle and crew scheduling," European Journal of Operational Research, Elsevier, vol. 16(2), pages 139-151, May.
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    Cited by:

    1. Kaul, Hemanshu & Rumpf, Adam, 2022. "A linear input dependence model for interdependent networks," European Journal of Operational Research, Elsevier, vol. 302(2), pages 781-797.
    2. Gensler, Sonja & Hinz, Oliver & Skiera, Bernd & Theysohn, Sven, 2012. "Willingness-to-pay estimation with choice-based conjoint analysis: Addressing extreme response behavior with individually adapted designs," European Journal of Operational Research, Elsevier, vol. 219(2), pages 368-378.
    3. Calvete, Herminia I., 2003. "Network simplex algorithm for the general equal flow problem," European Journal of Operational Research, Elsevier, vol. 150(3), pages 585-600, November.
    4. Giovanni Sechi & Paola Zuddas, 2008. "Multiperiod Hypergraph Models for Water Systems Optimization," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 22(3), pages 307-320, March.
    5. I N Kamal Abadi, 2007. "A new algorithm for minimizing makespan, C max, in blocking flow-shop problem through slowing down the operations," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 58(1), pages 134-140, January.
    6. Jordi Castro, 2007. "A Shortest-Paths Heuristic for Statistical Data Protection in Positive Tables," INFORMS Journal on Computing, INFORMS, vol. 19(4), pages 520-533, November.
    7. Giovanni M. Sechi & Riccardo Zucca, 2017. "A Cost-Simulation Approach to Finding Economic Optimality in Leakage Reduction for Complex Supply Systems," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 31(14), pages 4601-4615, November.
    8. Christina Büsing & Arie M. C. A. Koster & Sabrina Schmitz, 2022. "Robust minimum cost flow problem under consistent flow constraints," Annals of Operations Research, Springer, vol. 312(2), pages 691-722, May.
    9. David R. Morrison & Jason J. Sauppe & Sheldon H. Jacobson, 2013. "A Network Simplex Algorithm for the Equal Flow Problem on a Generalized Network," INFORMS Journal on Computing, INFORMS, vol. 25(1), pages 2-12, February.
    10. Jia Shu & Mabel C. Chou & Qizhang Liu & Chung-Piaw Teo & I-Lin Wang, 2013. "Models for Effective Deployment and Redistribution of Bicycles Within Public Bicycle-Sharing Systems," Operations Research, INFORMS, vol. 61(6), pages 1346-1359, December.
    11. Haiyan Lu & Enyu Yao & Liqun Qi, 2006. "Some further results on minimum distribution cost flow problems," Journal of Combinatorial Optimization, Springer, vol. 11(4), pages 351-371, June.
    12. Antonio Manca & Giovanni Sechi & Paola Zuddas, 2010. "Water Supply Network Optimisation Using Equal Flow Algorithms," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 24(13), pages 3665-3678, October.

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