A Network Simplex Algorithm for the Equal Flow Problem on a Generalized Network

A network simplex algorithm is described for the minimum-cost network flow problem on a generalized network with the additional constraint that there exist sets of arcs which must carry equal amounts of flow. This problem can be modeled as a linear programming problem and solved using the standard simplex algorithm. However, due to the structure of the problem, more efficient algorithms are possible that solve the problem by operating directly on the network itself. One such algorithm is described that leads to improved asymptotic performance per iteration over the standard simplex algorithm, as long as the number of side constraints is small relative to the size of the network. Computational results are given comparing this algorithm to CPLEX’s primal simplex solver on randomly-generated graphs.