The auction algorithm is an iterative method to find the optimal prices and an assignment that maximizes the net benefit, and is therefore the maximum weight matching (MWM).
An early auction algorithm was defined by Dimitri P. Bertsekas in 1988, with a later variation as an auction algorithm for shortest paths by Bertsekas in 1991. It is a simple algorithm for finding shortest paths in a directed graph. In the single origin/single destination case, the auction algorithm maintains a single path starting at the origin, which is then extended or contracted by a single node at each iteration. Simultaneously, at most one dual variable will be adjusted at each iteration, in order to either improve or maintain the value of a dual function. In the case of multiple origins, the auction algorithm is well-suited for parallel computation. The algorithm is closely related to auction algorithms for other network flow problems.
Auction algorithms for shortest hyperpath problems have been defined by De Leone and Pretolani in 1998. This is also a parallel auction algorithm for weighted bipartite matching, described by E. Jason Riedy in 2004.
Although in the auction algorithm, each iteration never decreases the total benefit (increases or remains the same), with the alternative Hungarian algorithm (from Kuhn, 1955; Munkres, 1957), each iteration always increases the total.
The auction algorithm of Bertsekas for finding shortest paths within a directed graph is reputed to perform very well on random graphs and on problems with few destinations.