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

摘要

We present the first polynomial-time approximation algorithms for {em single-minded envy-free profit-maximization problems}~cite{GuruswamiHKKKM05} with {em 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 {em any} LP-based $al$-approximation algorithm for the corresponding SWM problem as input and return a solution that achieves profit at least $OPT/O(alcdotlog u_{max})$, 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 $O(sqrt mlog u_{max})$ for the general single-minded envy-free problem; and $O(log u_{max})$ for the tollbooth and highway problems~cite{GuruswamiHKKKM05}, and the graph-vertex pricing problem~cite{BalcanB06} ($al=O(1)$ for all the corresponding SWM problems). Since $OPT$ is an upper bound on the maximum profit achievable by {em 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 theseproblems too. Our result also thus (constructively) establishes an upper bound of $O(alcdotlog u_{max})$ 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.