Efficiency of the Primal Network Simplex Algorithm for the Minimum-Cost Circulation Problem

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 -r -° on the number of pivots, for an nvertex network. This is the first known subexponential bound. The network simplex algorithm with this rule can be implemented. to run in n ogn) /2 + 0(l) tine." 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 as costdecreasing ones. For this algoithm we propose a pivot selection rule with a C" k bound of O(nm ..min {log(KCy, fi6g-l) 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 logn m min {(lognC), m logn)). This bound is competitive with those of previously known algorithms for the minimum-cost circulation problem. 1 esearch parially suppo.ted by the National Science Foundation, Grant No. DCR-8605961, and the Office of Naval Research, Contract No. N00014-7-K-0467. Efficiency of the Primal Network Simplex Algorithm for the Minimum-Cost Circulation Problem