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

• Authors: Maurice Cheung;Chaitanya Swamy
• Venue: FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2008

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 $\OPT/O(\al\cdot\log u_{\max})$, 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 $O(\sqrt m\log u_{\max})$ 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 $O(\al\cdot\log u_{\max})$ 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.