General Profit Scheduling and the Power of Migration on Heterogeneous Machines

Abstract

In this paper we consider the power of migration in heterogeneous machines settings and general profit scheduling. We begin by showing that on related machines or on related machines with restricted assignment that any migratory algorithm can be simulated by a non-migratory algorithm given 1+ε speed augmentation and O(1/ε) and O(1/ε2) machine augmentation, respectively, for any 0 < ε ≤ 1. Similar results were only known in the case of identical machines and our results effectively show that migration does not give too much additional power to an algorithm, even in heterogeneous environments. Our results are constructive and can be computed efficiently in the offline setting. We complement our result by showing that there exists migratory schedules on related machines which require Ω(1/ε) machine augmentation with (1+ε)-speed to be simulated by any non-migratory scheduler for any 0 < ε ≤ 1/2, showing that machine augmentation without speed augmentation is insufficient for a non-migratory scheduler to simulate a migratory scheduler. We then use these results to study general profit scheduling where a set of n jobs arrive over time online and every job i has a function gi(t) specifying the profit of completing job i at time t. The goal of the schedule is to maximize the total profit obtained. We give a (1+ε)-speed O(1/ε2)-competitive algorithm in the unrelated machines setting for any ε >0 when compared against a non-migratory adversary. Previous results were only known in the identical machines setting. As an example of the usefulness of the previous results on migration, they with the results on genial profit scheduling give a (1+ε)-speed O(1/ε4)-competitive algorithm for general profit scheduling when comparing against a migratory algorithm on related machines with restricted assignment for any ε >0.