# Online scheduling of jobs with favorite machines

## TL;DR

This paper studies online job scheduling on unrelated machines with favorite machines, providing tight bounds for greedy algorithms, characterizing optimal competitive ratios, and introducing a symmetric model with improved algorithms.

## Contribution

It introduces the favorite machine model, generalizes previous results, and develops new algorithms with tight bounds for online scheduling problems.

## Key findings

- Tight bounds on the greedy algorithm's competitive ratio.
- Characterization of the optimal competitive ratio for the favorite machine model.
- A 2.675-competitive algorithm for the symmetric favorite machine case.

## Abstract

This work introduces a natural variant of the online machine scheduling problem on unrelated machines, which we refer to as the favorite machine model. In this model, each job has a minimum processing time on a certain set of machines, called favorite machines, and some longer processing times on other machines. This type of costs (processing times) arise quite naturally in many practical problems. In the online version, jobs arrive one by one and must be allocated irrevocably upon each arrival without knowing the future jobs. We consider online algorithms for allocating jobs in order to minimize the makespan. We obtain tight bounds on the competitive ratio of the greedy algorithm and characterize the optimal competitive ratio for the favorite machine model. Our bounds generalize the previous results of the greedy algorithm and the optimal algorithm for the unrelated machines and the identical machines. We also study a further restriction of the model, called the symmetric favorite machine model, where the machines are partitioned equally into two groups and each job has one of the groups as favorite machines. We obtain a 2.675-competitive algorithm for this case, and the best possible algorithm for the two machines case.