On the simplex algorithm for networks and generalized networks

Abstract

We consider the simplex algorithm as applied to minimum cost network flows on a directed graph, G=(V, E). First we consider the strongly convergent pivot rule of Elam, Glover, and Klingman as applied to generalized networks. We show that this pivot rule is equivalent to Dantzig’s lexicographical rule in its choice of the variable to leave the basis. We also show the following monotonicity property that is satisfied by each basis B of a generalized network flow problem. If b′≤b≤b ^* and if l≤B ⁻¹ b′, B ⁻¹ b ^*≤u, then l≤B ⁻¹ b≤u; i.e., if a basis is feasible for b′ and b ^* then it is feasible for b. Next we consider Dantzig’s pivot rule of selecting the entering variable whose reduced cost is minimum and using lexicography to avoid cycling. We show that the maximum number of pivots using Dantzig’s pivot rule is O(|V|²|E| log |V|) when applied to either the assignment problem or the shortest path problem. Moreover, the maximum number of consecutive degenerate pivots for the minimum cost network flow problem is O(|V|²|E|log|V|).