Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions

We provide tight information-theoretic lower bounds for the welfare maximization problem in combinatorial auctions. In this problem, the goal is to partition m items among k bidders in a way that maximizes the sum of bidders' values for their allocated items. Bidders have complex preferences over items expressed by valuation functions that assign values to all subsets of items. We study the "black box" setting in which the auctioneer has oracle access to the valuation functions of the bidders. In particular, we explore the well-known value query model in which the permitted query to a valuation function is in the form of a subset of items, and the reply is the value assigned to that subset of items by the valuation function. We consider different classes of valuation functions: submodular,subadditive, and superadditive. For these classes, it has been shown that one can achieve approximation ratios of 1 - 1/e, 1/√m, and √log m/m, respectively, via a polynomial (in k and m) number of value queries. We prove that these approximation factors are essentially the best possible: For any fixed ε > 0, a (1 - 1/e + ε)-approximation for submodular valuations or an 1/m^1/2-ε-approximation for subadditive valuations would require exponentially many value queries, and a log^1+ε m/m-approximation for superadditive valuations would require a superpolynomial number of value queries.