Efficient Shortest Path Simplex Algorithms

We consider the specialization of the primal simplex algorithm to the problem of finding a tree of directed shortest paths from a given node to all other nodes in a network of n nodes or finding a directed cycle of negative length. Two efficient variants of this shortest path simplex algorithm are analyzed and shown to require at most (n − 1)(n − 2)/2 pivots and O(n 3 ) time.