(Equivalence of Two Forms of Reverse Auction) Show that the iteration of the Gauss-Seidel version of the reverse auction algorithm for the (symmetric) assignment problem can equivalently be described by the following iteration, which maintains an assignment and a pair (π, p) satisfying the -CS condition of Section 7.2.1 (cf. Definition 7.1):
Step 1: Choose an unassigned object j.
Step 2: Decrease pj to the highest level for which two or more persons will increase their profit by at least after assignment to j, that is, set pj to the highest level for which 
for at least two persons i, where πi is the profit of i at the start of the iteration.
Step 3: From the persons in Step 2, assign to j a person ij that experiences maximum profit increase after assignment to j, and cancel the prior assignment of ij if he or she was assigned at the start of the iteration. Set the profit of ij to