New efficient shortest path simplex algorithm: pseudo permanent labels instead of permanent labels

We introduce a new network simplex pivot rule for the shortest path simplex algorithm. This new pivot rule chooses a subset of non-basic arcs to enter into the basis simultaneously. We call to this operation multiple pivot. We show that a shortest path simplex algorithm with this pivot rule makes O(n) multiple pivots and runs in O(nm) time. This new pivot rule is based on the new concept of pseudo permanently labeled node that it can be without effort adapted to design a new label-correcting algorithm that runs in O(nm). Moreover, this concept let us to introduce a new rule to identify negative cycles. Finally, we compare the network simplex algorithm with multiple pivots with other previously proposed efficient network simplex algorithm in a computational experiment.