Optimal clearing of supply/demand curves

Markets are important coordination mechanisms for multiagent systems, and market clearing has become, a key application area of algorithms. We study optimal clearing in the ubiquitous setting where there are multiple indistinguishable units for sale. The sellers and buyers express their bids via supply and demand curves. Discriminatory pricing leads to greater profit for the party who runs the market than nondiscriminatory pricing. We show that this comes at the cost of computation complexity. For piecewise linear curves we present a fast polynomial-time algorithm for nondiscriminatory clearing, and show that discriminatory clearing is NP-complete (even in a very special case). We then show that in the more restricted setting of linear curves, even discriminatory markets can be cleared fast in polynomial time. Our derivations also uncover the elegant fact that to obtain the optimal discriminatory solution, each buyer's (seller's) price is incremented (decremented) equally from that agent's price in the quantity-unconstrained solution.