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Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

By Parinya Chalermsook, Julia Chuzhoy, Sampath Kannan and Sanjeev Khanna

Abstract

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
OAI identifier: oai:CiteSeerX.psu:10.1.1.308.6231
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