Winner determination in combinatorial auction generalizations

Combinatorial markets where bids can be submitted on bundles of items can be economically desirable coordination mechanisms in multiagent systems where the items exhibit complementarity and substitutability. There has been a surge of research on winner determination in combinatorial auctions. In this paper we study a wider range of combinatorial market designs: auctions, reverse auctions, and exchanges, with one or multiple units of each item, with and without free disposal. We first theoretically characterize the complexity of finding a feasible, approximate, or optimal solution. Reverse auctions with free disposal can be approximated (even in the multi-unit case), although auctions cannot. When XOR-constraints between bids are allowed (to express substitutability), the hardness turns the other way around: even finding a feasible solution for a reverse auction or exchanges is &Ngr;&Pgr;-complete, while in auctions that is trivial. Finally, in all of the cases without free disposal, even finding a feasible solution is &Ngr;&Pgr;-complete.We then ran experiments on known benchmarks as well as ones which we introduced, to study the complexity of the market variants in practice. Cases with free disposal tended to be easier than ones without. On many distributions, reverse auctions with free disposal were easier than auctions with free disposal---as the approximability would suggest---but interestingly, on one of the most realistic distributions they were harder. Single-unit exchanges were easy, but multi-unit exchanges were extremely hard.