A Polynomial Time Primal Network Simplex Algorithm for Minimum Cost Flows (An Extended Abstract)

Developing a polynomial time 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(n 2m log nC, n2m 2 log n)) time, where n is the number of nodes in the network, m is the number of arcs, and C denotes the maximum absolute arc costs if arc costs are integer and 0 otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the "premultiplier algorithm." A vector X of node potentials is called a vector of premultipliers with respect to a rooted tree if each arc directed towards the root has a non-positive reduced cost and each arc directed away from the root has a non-negative reduced cost. 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).