(Noninteger Problem Data) Verify that the primal-dual method terminates even when the arc costs are noninteger. (Note, however, that the arc flow bounds must still be integer; the max- flow example of Exercise 3.7 in Chapter 3 applies to the primal-dual method as well, in view of the relation described in Exercise 6.2.) Modify the primal-dual method so that augmenting paths have as few arcs as possible. Show that with this modification, the arc flow bounds need not be integer for the method to terminate. How should the sequential shortest path method be modified so that it terminates even when the problem data are not integer?
Exercise 6.2.) Relation of Primal-Dual and Dijkstra) Consider the shortest path problem with node 1 being the origin and all other nodes being destinations. Formulate this problem as a minimum cost flow problem with the origin having supply N - 1 and all destinations having demand 1. Assume that all arc lengths are nonnegative. Start with all flows and prices equal to zero, and apply the primal-dual method. Show that the method is equivalent to Dijkstra's algorithm. In particular, each augmentation uses a shortest path from the origin to some destination, the augmentations are done in the order of the destinations' proximity to the origin, and upon termination, p1 -pi gives the shortest distance from 1 to each destination i that can be reached from the origin via a forward path.\