(Relation of Path Construction and Assignment) The purpose of this exercise (from Bertsekas [1995c]) is to show the connection of the path construction algorithm of Section 3.3.1 with the assignment auction algorithm of Section 1.3.3.
(a) Show that the path construction problem can be converted into the problem of finding a solution of a certain assignment problem with all arc values equal to 0, as shown by example in Fig. 3.13. In particular, a forward path of a directed graph G that starts at node s and ends at node t corresponds to a feasible solution of the assignment problem, and conversely.
(b) Show how to relate the node prices in the path construction algorithm with the object prices of the assignment problem, so that if we apply the auction algorithm with = 1, the sequence of generated prices and assignments corresponds to the sequence of generated prices and paths by the path construction algorithm.