A polynomial time primal network simplex algorithm for minimum cost flows

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n²m log nC,n²m² log n)) time, where n is the number of nodes in the network, m is the nuber of arcs, and C denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the ‘premultiplier algorithm’. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm log nC,nm² 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 O(nm log n).