Auction/Belief Propagation Algorithms for Constrained Assignment Problem

In this paper, we investigate the constrained assignment problem: a set of offers are to be assigned to a set of customers. There are constraints on both the number of available copies for each offer and the number of offers one customer can get. To measure the assignment gain, we have a score for each customer-offer pair, quantifying how beneficial it is for assigning a customer an offer. Additionally, a customer can get at most one copy of the same offer and at most one offer from the same category (an offer is associated with a category in a taxonomy, for example bakery, dairy, and so on). The objective is to optimize the assignment so that the sum of scores (global benefits) are maximized. We developed an auction algorithm for this problem and proved both its correctness and convergence. To show its effectiveness and efficiency, we compared it with heuristic algorithms and one minimum cost flow algorithm (network simplex). To show its scalability, we ran test cases of size up to 200 offers × 3,000,000 customers on a 16GB machine, where most of the linear/integer programming tools would fail under this setting. Finally, we transformed the auction algorithm into an equivalent belief propagation algorithm, and provided another convergence case for belief propagation on a loopy graph with node constraints.