# 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
as

File URL: http://dx.doi.org/10.1287/mnsc.45.10.1440

## References listed on IDEAS

as
1. 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.
2. 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.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. 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.
2. 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.
3. 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.
4. repec:pal:jorsoc:v:58:y:2007:i:1:d:10.1057_palgrave.jors.2602099 is not listed on IDEAS
5. 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.

### Keywords

minimum cost flow problem; network simplex algorithm; scaling algorithm; parametric programming; network flows;

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