Auction algorithm for the assignment problem

The maximum bipartite matching is a well-known problem in graph theory, operations research, and economics. Also known by the name assignment problem, it models a marketplace with buyers and items, where every buyer has a valuation for each item, and we want to match (assign) each buyer to an item, with no item being shared. For social utility, an assignment that maximizes the total valuation is ideal. In graph theory, the Hungarian algorithm by Kuhn produces a matching in polynomial time maximizing the total weight of the edges. Independently, Bertsekas from operations research and Demange, Gale, and Sotomayor from the economics perspective both use an underlying auction to solve the same problem. […]