A QPTAS for -Envy-Free Profit-Maximizing Pricing on Line Graphs

Abstract

We consider the problem of pricing edges of a line graph so as to maximize the profit made from selling intervals to single-minded customers. An instance is given by a set E of n edges with a limited supply for each edge, and a set of m clients, where each client j specifies one interval of E she is interested in and a budget B _j which is the maximum price she is willing to pay for that interval. An envy-free pricing is one in which every customer is allocated (possibly empty) interval maximizing her utility. Recently, Grandoni and Rothvoss (SODA 2011) gave a polynomial-time approximation scheme (PTAS) for the unlimited supply case with running time . By utilizing the known hierarchical decomposition of doubling metrics, we give a PTAS with running time . We then consider the limited supply case, and the notion of -envy-free pricing in which a customer gets an allocation maximizing her utility within an additive error of . For this case we develop an approximation scheme with running time , where $H_e=\frac{B_{\max}(e)}{B_{\min}(e)}$ is the maximum ratio of the budgets of any two customers demanding edge e. This yields a PTAS in the uniform budget case, and a quasi-PTAS for the general case.