Approximation Algorithms for Single-minded Envy-free Profit-maximization Problems with Limited Supply

Abstract

We present the first polynomial-time approximation algorithms for single-minded envy-free profit-maximization problems [13] with limited supply. Our algorithms return a pricing scheme and a subset of customers that are designated the winners, which satisfy the envy-freeness constraint, whereas in our analyses, we compare the profit of our solution against the optimal value of the corresponding social-welfare-maximization (SWM) problem of finding a winner-set with maximum total value. Our algorithms take any LP-based α-approximation algorithm for the corresponding SWM problem as input and return a solution that achieves profit at least OPT /O(α · log u max ), where OPT is the optimal value of the SWM problem, and u max is the maximum supply of an item. This immediately yields approximation guarantees of O( √ m log u max ) for the general single-minded envy-free problem; and O(log u max ) for the tollbooth and highway problems [13], and the graph-vertex pricing problem [3] (α = O(1) for all the corresponding SWM problems). Since OPT is an upper bound on the maximum profit achievable by any solution (i.e., irrespective of whether the solution satisfies the envy-freeness constraint), our results directly carry over to the non-envy-free versions of these problems too. Our result also thus (constructively) establishes an upper bound of O(α · log u max ) on the ratio of (i) the optimum value of the profit-maximization problem and OPT ; and (ii) the optimum profit achievable with and without the constraint of envy-freeness.

Improved Hardness Results for Profit Maximization Pricing Problems with Unlimited Supply

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Online Pricing for Multi-type of Items

“…Previous work has been focused on two supply models: the unlimited supply model [1,2,6,10,12] where the number of each type of item is unbounded and the limited supply model [1, 3-5, 7, 11, 13, 14] where the number of each type of item is bounded by some value. As for the users, there are several users' behaviors studied, including single-minded [7-10, 12, 14] (each user is only interested in a particular set of items), unit-demand [2-6, 12, 14] (each user will buy at most one item in total) and envy free [1,5,7,10,12] (after the assignment, no user would prefer to be assigned a different set of items with the designated prices, loosely speaking, each user is happy with his/her purchase). Most of the previous studies have considered a combination of the above scenarios (e.g.…”

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