Developing a polynomial time primal network simplex algorithm for the mininumn cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n ml. log nC,n -l log n)) time, where n is the number of nodes in the network, nl is the number of arcs, and C denotes the maximum absolute arc costs if arc costs are integer and x otherwise. W'e first introduce a pseudopolynomial variant of the network simplex algorithni called the 'premultiplier algorithm". We then develop a cost-scaling version of the premultiplier , Israel algorithm that solves the minimum cost flow problem in O(min(nni log nC, nm 2 log n)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is 0(nn log n). © 1997 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. Kcv'orsc.s: N'lininlLmun cost flows; Network simplex; Polynomial time; Simplex algorithin Premulitipterss

@inproceedings{OrlinAPT,
title={A polynomial time primal network simplex , USA anston , algorithm for minimum cost flows},
author={James B. Orlin}
}