Online Makespan Scheduling with Job Migration on Uniform Machines

Abstract

In the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to change the assignment of up to k jobs at the end for some limited number k. For m identical machines, Albers and Hellwig (Algorithmica 79(2):598–623, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and $$\\approx 1.4659$$\n \n ≈\n 1.4659\n \n . They show that $$k = O(m)$$\n \n k\n =\n O\n (\n m\n )\n \n is sufficient to achieve this bound and no $$k = o(n)$$\n \n k\n =\n o\n (\n n\n )\n \n can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a $$\\delta = \\varTheta (1)$$\n \n δ\n =\n Θ\n (\n 1\n )\n \n such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than $$1.4659 + \\delta $$\n \n 1.4659\n +\n δ\n \n with $$k = o(n)$$\n \n k\n =\n o\n (\n n\n )\n \n . We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and $$\\approx 1.7992$$\n \n ≈\n 1.7992\n \n with $$k = O(m)$$\n \n k\n =\n O\n (\n m\n )\n \n . We also show that $$k = \\varOmega (m)$$\n \n k\n =\n Ω\n (\n m\n )\n \n is necessary to achieve a competitive ratio below 2. Our algorithm is based on maintaining a specific imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines.