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Algorithms for the Simple Equal Flow Problem

  • 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)

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    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.

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    File URL: http://dx.doi.org/10.1287/mnsc.45.10.1440
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    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 45 (1999)
    Issue (Month): 10 (October)
    Pages: 1440-1455

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    Handle: RePEc:inm:ormnsc:v:45:y:1999:i:10:p:1440-1455
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