The Scaling Network Simplex Algorithm

Abstract

In this paper, we present a new primal simplex pivot rule and analyze the worst case complexity of the resulting simplex algorithm for the minimum cost flow, the assignment, and the shortest path problems. We consider networks with n nodes, m arcs, integral arc capacities bounded by an integer number U, and integral arc costs whose magnitudes are bounded by an integer number C. Our pivot rule may be regarded as a scaling version of Dantzig's pivot rule. Our pivot rule defines a threshold value Δ, which is initially at most 2C, and the rule permits any arc with a violation of at least Δ/2 to be the editing variable. We select the leaving arc so that strong feasibility of the basis is maintained. When there is no arc satisfying this rule, then we replace Δ by Δ/2 and repeat the process. The algorithm terminates when Δ < 1. We show that the simplex algorithm using this rule performs O(nmU log C) pivots and can be implemented to run in O(m2U log C) time. Specializing these results for the assignment and shortest path problems we show that the simplex algorithm solves these problems in O(n2 log C) pivots and O(nm log C) time.