Mathematical Equivalence of the Auction Algorithm for Assignment and the ∊-Relaxation (Preflow-Push) Method for Min Cost Flow

It is well known that the linear minimum cost flow network problem can be converted to an equivalent assignment problem. Here we give a simple proof that when the auction algorithm is applied to this equivalent problem, one obtains the generic form of the ∊-relaxation method, and as a special case, the Goldberg-Tarjan preflow-push max-flow algorithm. The reverse equivalence is already known, that is, if we view the assignment problem as a special case of a minimum cost flow problem and we apply the ∊-relaxation method with some special rules for choosing the node to iterate on, we obtain the auction algorithm. Thus, the two methods are mathematically equivalent.