Online Load Balancing for Related Machines

TL;DR

This paper provides a comprehensive solution to online load balancing for related machines, achieving constant competitive algorithms for scalar and vector jobs across various norms, and clarifies the complexity differences in vector scheduling variants.

Contribution

It introduces the first constant competitive algorithms for $ ext{ell}_q$-norms on related machines and characterizes the complexity of vector scheduling variants, contrasting related and identical machine models.

Findings

Constant competitive algorithms for scalar $ ext{ell}_q$-norm scheduling.

Sharp contrast between vector scheduling variants, relating to unrelated and identical machines.

Extension of results to arbitrary $ ext{ell}_q$-norms for vector loads.

Abstract

In the load balancing problem, introduced by Graham in the 1960s (SIAM J. of Appl. Math. 1966, 1969), jobs arriving online have to be assigned to machines so to minimize an objective defined on machine loads. A long line of work has addressed this problem for both the makespan norm and arbitrary $\ell_q$-norms of machine loads. Recent literature (e.g., Azar et al., STOC 2013; Im et al., FOCS 2015) has further expanded the scope of this problem to vector loads, to capture jobs with multi-dimensional resource requirements in applications such as data centers. In this paper, we completely resolve the job scheduling problem for both scalar and vector jobs on related machines, i.e., where each machine has a given speed and the time taken to process a job is inversely proportional to the speed of the machine it is assigned on. We show the following results. For scalar scheduling, we give a constant competitive algorithm for optimizing any $\ell_q$-norm for related machines. The only previously known result was for the makespan norm. For vector scheduling, there are two natural variants for vector scheduling, depending on whether the speed of a machine is dimension-dependent or not. We show a sharp contrast between these two variants, proving that they are respectively equivalent to unrelated machines and identical machines for the makespan norm. We also extend these results to arbitrary $\ell_q$-norms of the machine loads. No previous results were known for vector scheduling on related machines.