A genuinely polynomial primal simplex algorithm for the assignment problem

AbstractWe present a primal simplex algorithm that solves the assignment problem in 12n(n+3)−4 pivots. Starting with a problem of size 1, we sequentially solve problems of size 2,3,4,…,n. The algorithm utilizes degeneracy by working with strongly feasible trees and employs Dantzig's rule for entering edges for the subproblem. The number of nondegenerate simplex pivots is bounded by n−1. The number of consecutive degenerate simplex pivots is bounded by 12(n−2)(n+1). All three bounds are sharp. The algorithm can be implemented to run in O(n3) time for dense graphs. For sparse graphs, using state of the art data structures, it runs in O(n2 log n+nm) time, where the bipartite graph has 2n nodes and m edges