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We consider profit maximization pricing problems, where we are given a set of m customers and a set of n items. Each customer c is associated with a subset Sc ⊆ [n] of items of interest, together with a budget Bc, and we assume that there is an unlimited supply of each item. Once the prices are fixed for all items, each customer c buys a subset of items in Sc, according to its buying rule. The goal is to set the item prices so as to maximize the total profit. We study the unit-demand min-buying pricing (UDPMIN) and the single-minded pricing (SMP) problems. In the former problem, each customer c buys the cheapest item i ∈ Sc, if its price is no higher than the budget Bc, and buys nothing otherwise. In the latter problem, each customer c buys the whole set Sc if its total price is at most Bc, and buys nothing otherwise. Both problems are known to admit O(min {log(m + n), n})-approximation algorithms. We prove that they are log 1−ɛ (m+n) hard to approximate for any constant ɛ, unless NP ⊆ DTIME(n logδ n where δ is a constant depending on ɛ. Restricting our attention to approximation factors depending only on n, we show that these problems are 2log1−δ n-hard to approximate for any δ> 0 unless NP ⊆ ZPTIME(nlogδ ′ n ′), where δ is some constant depending on δ. We also prove tha

Year: 2013

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