Winner Determination Algorithms for Combinatorial Auctions with Sub-cardinality Constraints

Abstract

We examine the winner determination problem for combinatorial auctions with sub-cardinality constraints (WDP-SC). In this type of auction, bidders submit bids for packages of items of interest together with a specific number of items they want. All items in a package are equally acceptable, but each bidder wants only the specified number of items in their package and not necessarily all. This type of auction is particularly relevant in cases where bidders are interested in items in close proximity to one another and view items as largely substitutable. We show that WDP-SC is NP-hard. We then develop an integer programming (IP) formulation and two approximation algorithms for solving WDP-SC: one based on simulated annealing (SA) and one based on the Metaheuristic for Randomized Priority Search (Meta-RaPS). We assess the performance of these three methods through computational tests performed on sets of large benchmark problems (ranging from 1000 to 10,000 bids) generated from four different distributions: two developed specifically to simulate proximity concerns, and two others previously used in the literature. IP produced the best solutions for 1000-bid problems, but in many cases SA and Meta-RaPS produced better solutions than IP on larger problems. Both IP and Meta-RaPS appear to be robust solution methods for this problem and consistently produced high-quality solutions across all problem sizes and distributions.