Mechanisms for multi-unit combinatorial auctions with a few distinct goods

We design and analyze deterministic truthful approximation mechanisms for multi-unit Combinatorial Auctions with only a constant number of distinct goods, each in arbitrary limited supply. Prospective buyers (bidders) have preferences over multisets of items, i.e. for more than one unit per distinct good. Our objective is to determine allocations of multisets that maximize the Social Welfare. Despite the recent theoretical advances on the design of truthful Combinatorial Auctions (for several distinct goods) and multi-unit auctions (for a single good), results for the combined setting are much scarser. Our main results are for multi-minded and submodular bidders. In the first setting each bidder has a positive value for being allocated one multiset from a prespecified demand set of alternatives. In the second setting each bidder is associated to a submodular valuation function that defines his value for the multiset he is allocated. For multi-minded bidders we design a truthful FPTAS that fully optimizes the Social Welfare, while violating the supply constraints on goods within factor (1+e) for any fixed e > 0 (i.e., the approximation applies to the constraints and not to the Social Welfare). This result is best possible, in that full optimization is impossible without violating the supply constraints. It also improves significantly upon a related result of Grandoni et al. [SODA 2010]. For submodular bidders we extend a general technique by Dobzinski and Nisan [JAIR, 2010] for multi-unit auctions, to the case of multiple distinct goods. We use this extension to obtain a PTAS that approximates the optimum Social Welfare within factor (1+e) for any fixed e > 0, without violating the supply constraints. This result is best possible as well. Our allocation algorithms are Maximum-in-Range and yield truthful mechanisms when paired with Vickrey-Clarke-Groves payments.