Symmetry Exploitation for Online Machine Covering with Bounded Migration

Waldo Gálvez,José A Soto,José Verschae

doi:10.1145/3397535

Abstract

Online models that allow recourse can be highly effective in situations where classical online models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful structural properties for online packing and covering problems, we design first a simple (1.7 + ε)-competitive algorithm using a migration factor of O(1/ε), which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved (4/3+ε)-competitive algorithm using a migration factor of Ō (1/ε 3 ). At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure.

Highlights

We consider a fundamental load balancing problem where n jobs need to be assigned to m identical parallel machines
Given an assignment of jobs, the load of a machine is the sum of the processing times of jobs assigned to it
The machine covering problem asks for an assignment of jobs to machines maximizing the load of the least loaded machine

Summary

Introduction

We consider a fundamental load balancing problem where n jobs need to be assigned to m identical parallel machines. In order to show the usefulness of the rounding procedure, we first present a simple (1.7 + ε)-competitive algorithm using a migration factor of O(1/ε) This algorithm maintains through the arrival of new jobs a locally optimal solution with respect to Jump for large jobs and a greedy assignment for small jobs on top of that. The analysis of the algorithm will rely on monotonicity properties implied by LPT which, coupled with rounding, implies that for every job size the increase in the number of machines with different assignments (w.r.t the solution of the previous time step) is constant. This yields a migration factor that only grows polynomially in 1/ε. We defer most of the proofs to the full version [9]

Preliminaries

Algorithms with robust structure

Rounding procedure