Symmetry exploitation for Online Machine Covering with Bounded Migration

TL;DR

This paper introduces new online algorithms for machine covering with bounded migration, achieving improved competitive ratios by exploiting symmetry and structural properties in job assignment.

Contribution

It presents two algorithms with better competitive ratios for online machine covering, utilizing symmetry and rounding techniques to control migration.

Findings

Achieved a (1.7 + ε)-competitive algorithm with O(1/ε) migration

Developed a (4/3 + ε)-competitive algorithm with Õ(1/ε^3) migration

Exploited symmetry in rounding to limit job reassignment during online processing

Abstract

Online models that allow recourse are highly effective in situations where classical 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 $(1.7 + \varepsilon)$-competitive algorithm using a migration factor of $O(1/\varepsilon)$ 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 $(4/3+\varepsilon)$-competitive algorithm using a migration factor of $\tilde{O}(1/\varepsilon^3)$. 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.