On hardness of pricing items for single-minded bidders

We consider the following item pricing problem which has received much attention recently. A seller has an infinite numbers of copies of n items. There are m buyers, each with a budget and an intention to buy a fixed subset of items. Given prices on the items, each buyer buys his subset of items, at the given prices, provided the total price of the subset is at most his budget. The objective of the seller is to determine the prices such that her total profit is maximized. In this paper, we focus on the case where the buyers are interested in subsets of size at most two. This special case is known to be APX-hard (Guruswami et al [1]). The best known approximation algorithm, by Balcan and Blum, gives a 4-approximation [2]. We show that there is indeed a gap of 4 for the combinatorial upper bound used in their analysis. We further show that a natural linear programming relaxation of this problem has an integrality gap of 4, even in this special case. Then we prove that the problem is NP-hard to approximate within a factor of 2 assuming the Unique Games Conjecture; and it is unconditionally NP-hard to approximate within a factor 17/16. Finally, we extend the APX-hardness of the problem to the special case in which the graph formed by items as vertices and buyers as edges is bipartite. We hope that our techniques will be helpful for obtaining stronger hardness of approximation bounds for this problem.