Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

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 {log(m + n), n})-approximation algorithms. We prove that they are log 1− (m+n) hard to approximate for any constant , unless NP ⊆ DTIME(n log δ n ), where δ is a constant depending on . Restricting our attention to approximation factors depending only on n, we show that these problems are 2 log 1−δ n -hard to approximate for any δ > 0 unless NP ⊆ ZPTIME(n log δ 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− -hard to approximate for any constant . 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 n)-hardness of approximation, assuming NP ⊆ BPTIME(2 n δ ), for any constant δ, and some constant depending on δ.

Online pricing for multi-type of items

“…They gave an approximation algorithm which achieves expected revenue within an O (log n + log m) factor of the total social welfare. This approximation ratio of O (log n + log m) was proved to be tight by Briest [6] and Chalermsook et al [8]. When m different types with unlimited supplies are for sale, and users are single-minded and each wants at most k types of items, Briest et al [7] gave an algorithm with approximation ratio O (k 2 ), which was later improved to O (k) by Balcan et al [2].…”

“….} with probability Pr[κ = i] = q i ; 2 Set the unit price τ := 2 κ ; 3 Choose an integer γ from the set {1, √ k + 1} uniformly at random as the bundle size threshold; 4 For each 1 ≤ j ≤ k, set R j := m j as the quantity of the remaining available items of type j; 5 while a new user i arrives do 6 if |I i | ≥ γ then 7 M := min j∈Ii R j ; 8 Let y be the largest quantity that i is willing to buy given the unit price τ , i.e., the largest y ∈ N with ϕ i (y) ≥ τ ; 9 Sell x i := min(M, y) bundles to the user i at unit price p i := τ ;…”

“…There have been extensive studies on the offline pricing problem [14,6,8,7,2]. Guruswami et al [14] studied the problem with unlimited supplies of m different types of items and n single-minded users.…”

“…For each ≥ 0, let z be the largest amount of bundles that the user is willing to buy given price 2 and satisfying z ≤ m; 7 For each ≥ 0, let y = min{z , min j∈I {σ ,t j }}; 8 Let κ = arg max y · 2 such that y > 0; 9 if no such κ exists then 10 Assign 0 bundles to the user; stages 0 can be only sold to users with bundle size |I i | < √ k + 1 and items in stages 1 can be only sold to users with |I i | ≥ √ k + 1.…”

Untitled

“…The hardness of approximation of GVP was recently shown to be 2 − ε, assuming the Unique Games Conjecture [22]. Even more recently, the hardness of k 1−ε for SMP was shown by [12] (building on [7,11,10]), assuming P = NP where k is the size of the largest hyperedge.…”