On the hardness of pricing loss-leaders

Consider the problem of pricing n items under an unlimited supply with m buyers. Each buyer is interested in a bundle of at most k of the items. These buyers are single minded, which means each of them has a budget and they will either buy all the items if the total price is within their budget or they will buy none of the items. The goal is to price each item with profit margin p1, p2, ..., pn so as to maximize the overall profit. When k = 2, such a problem is called the graph-vertex-pricing problem. Another special case of the problem is the highwaypricing problem when the items (toll-booths) are arranged linearly on a line and each buyer (as a driver) is interested in paying for a path that consists of consecutive items. The goal again is to price the items (tolls) so as to maximize the total profits. There is an O(k)-approximation algorithm by [BB06] when the price on each item must be above its margin cost; i.e., pi > 0 for every i ∈ [n]. As for the highway problem, a PTAS is shown in [GR11]. We investigate the above problem when the seller is allowed to price some of the items below their margin cost. It is shown in [BB06, BBCH07] that by pricing some of the items below cost, the maximum profit can increase by a factor of Ω(log n). These items sold at low prices to stimulate other profitable sales are called “loss leaders”. Given the possibility of making more profit, understanding the approximability of pricing loss leaders for graphvertex-pricing, highway-pricing as well as the general item pricing problem are formulated as open problems in [BB06,BBCH07]. In this paper, we obtain strong hardness of approximation result for the problem of pricing loss leaders. First we show that it is NP-hard to get better than O(log log log n)-approximation when k ≥ 3. This improves a previous super-constant hardness result assuming the Unique Games Conjecture [Wu11]. In addition, we show a super-constant Unique-Games hardness for the highway-pricing problem as well as the graph-vertex-pricing problem.