# Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

• Parinya Chalermsook
• Julia Chuzhoy
• Sampath Kannan
• Sanjeev Khanna
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)

## 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 S c  ⊆ [n] of items of interest, together with a budget B c , 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 S c , 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 (UDP MIN ) and the single-minded pricing (SMP) problems. In the former problem, each customer c buys the cheapest item i ∈ S c , if its price is no higher than the budget B c , and buys nothing otherwise. In the latter problem, each customer c buys the whole set S c if its total price is at most B c , and buys nothing otherwise. Both problems are known to admit $$O(\min \left\{ \log (m+n), n \right\})$$-approximation algorithms. We prove that they are log1 − ε (m + n) hard to approximate for any constant ε, unless $$\mbox{\sf NP} \subseteq{\sf DTIME}(n^{\log^{\delta} n})$$, where δ is a constant depending on ε. Restricting our attention to approximation factors depending only on n, we show that these problems are $$2^{\log^{1-\delta} n}$$-hard to approximate for any δ > 0 unless $$\mbox{\sf NP} \subseteq{\sf ZPTIME}(n^{\log^{\delta'} n})$$, where δ′ is some constant depending on δ. We also prove that restricted versions of UDP MIN and SMP, where the sizes of the sets S c are bounded by k, are k 1/2 − ε -hard to approximate for any constant ε.

We then turn to the Tollbooth Pricing problem, a special case of SMP, where each item corresponds to an edge in the input graph, and each set S c is a simple path in the graph. We show that Tollbooth Pricing is at least as hard to approximate as the Unique Coverage problem, thus obtaining an Ω(log ε n)-hardness of approximation, assuming $$\mbox{\sf NP}\not\subseteq \mbox{\sf BPTIME}(2^{n^{\delta}})$$, for any constant δ, and some constant ε depending on δ.

## Keywords

Approximation Algorithm Approximation Factor Input Graph Price Problem Simple Path
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2012

## Authors and Affiliations

• Parinya Chalermsook
• 1
• 2
• Julia Chuzhoy
• 3
• Sampath Kannan
• 4
• Sanjeev Khanna
• 4
1. 1.University of ChicagoChicagoUSA
2. 2.IDSIALuganoSwitzerland
3. 3.Toyota Technological InstituteChicagoUSA
4. 4.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphiaUSA