Published on *Computer Science Department at Princeton University* (https://www.cs.princeton.edu)

We study the number of pivots required by the primal network simplex algorithm to solve the minimum-cost circulation problem. We propose a pivot selection rule with a bound of n(log n)/2+0(1) on the number of pivots, for an n-vertex network. This is the first known subexponential bound. The network simplex

algorithm with this rule can be implemented to run in n(log n)/2+0(1) time. In the special case of planar graphs, we obtain a polynomial bound on the number of pivots and the running time. We also consider the relaxation of the network simplex algorithm in which cost-increasing pivots are allowed as well cost-decreasing ones. For this algorithm we propose a pivot selection rule with a bound of O(nm * min{log(nC), m log n}) on the number of pivots, for a network with n vertices, m arcs, and integer arc costs bounded in magnitude by C. The total running time is O(nm log n * min{(log nC), m log n}). This bound is competitive with those of previously known algorithms for the minimum-cost circulation problem.

**Links**

[1] https://www.cs.princeton.edu/research/techreps/author/384