Parallel solutions for ordinal scheduling with a small number of machines

TL;DR

This paper investigates parallel algorithms for ordinal makespan scheduling on small numbers of identical machines, providing tight bounds and improved competitive ratios by considering pairs of solutions.

Contribution

It introduces a new approach for analyzing pairs of solutions, achieving better competitive ratios for two to five machines in ordinal scheduling.

Findings

Tight bounds for two and three machines.

Algorithms with better ratios for four and five machines.

A novel method considering pairs of solutions for analysis.

Abstract

We study ordinal makespan scheduling on small numbers of identical machines, with respect to two parallel solutions. In ordinal scheduling, it is known that jobs are sorted by non-increasing sizes, but the specific sizes are not known in advance. For problems with two parallel solutions, it is required to design two solutions, and the performance of algorithm is tested for each input using the best solution of the two. We find tight results for makespan minimization on two and three machines, and algorithms that have strictly better competitive ratios than the best possible algorithm with a single solution also for four and five machines. To prove upper bounds, we use a new approach of considering pairs of machines from the two solutions.