Demand Queries with Preprocessing

Given a set of items and a submodular set-function f that determines the value of every subset of items, a demand query assigns prices to the items, and the desired answer is a set S of items that maximizes the profit, namely, the value of S minus its price. The use of demand queries is well motivated in the context of combinatorial auctions. However, answering a demand query (even approximately) is NP-hard. We consider the question of whether exponential time preprocessing of f prior to receiving the demand query can help in later answering demand queries in polynomial time. We design a preprocessing algorithm that leads to approximation ratios that are NP-hard to achieve without preprocessing. We also prove that there are limitations to the approximation ratios achievable after preprocessing, unless NP subset of P/poly.