Linear, Exponential, but Nothing Else

Abstract

We consider two seemingly unrelated resource allocation problems and show that they share a deep structural property. In the first problem, we schedule jobs on a single machine to minimize the sum of the jobs’ cost where each job’s cost is determined by a job-specific function of its completion time. In the second problem, we consider weighted congestion games and are interested in the existence of pure Nash equilibria. We show that the classes of delay cost functions for which the scheduling problem admits a priority rule are exactly the classes of resource cost functions that guarantee the existence of a pure Nash equilibrium in weighted congestion games. These classes of cost functions are those that contain only linear functions or exponential functions, but not both.