Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

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


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.


Approximation Algorithm Approximation Factor Input Graph Price Problem Simple Path 
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Copyright information

© 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

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