Market Clearability

Market mechanisms play a central role in AI as a coordination tool in multiagent systems and as an application area for algorithm design. Mechanisms where buyers are directly cleared with sellers, and thus do not require an external liquidity provider, are highly desirable for electronic marketplaces for several reasons. In this paper we study the inherent complexity of, and design algorithms for, clearing auctions and reverse auctions with multiple indistinguishable units for sale. We consider settings where bidders express their preferences via price-quantity curves, and settings where the bids are price-quantity pairs. We show that markets with piecewise linear supply/demand curves and non-discriminatory pricing can always be cleared in polynomial time. Surprisingly, if discriminatory pricing is used to clear the market, the problem becomes NP-Complete (even for step function curves). If the price-quantity curves are all linear, then, in most variants, the problem admits a poly-time solution even for discriminatory pricing. When bidders express their preferences with price-quantity pairs, the problem is NP-Complete, but solvable in pseudo-polynomial time. With free disposal, the problem admits a poly-time approximation scheme, but no such approximation scheme is possible without free disposal. We also present pseudo-polynomial algorithms for XOR bids and or-of-xors bids, and analyze the approximability.