## Abstract

Makespan minimization on parallel identical machines is a classical and intensively studied problem in scheduling, and a classic example for online algorithm analysis with Graham’s famous list scheduling algorithm dating back to the 1960s. In this problem, jobs arrive over a list and upon an arrival, the algorithm needs to assign the job to a machine. The goal is to minimize the makespan, that is, the maximum machine load. In this paper, we consider the variant with an additional cardinality constraint: The algorithm may assign at most k jobs to each machine where k is part of the input. While the offline (strongly NP-hard) variant of cardinality constrained scheduling is well understood and an EPTAS exists here, no non-trivial results are known for the online variant. We fill this gap by making a comprehensive study of various different online models. First, we show that there is a constant competitive algorithm for the problem and further, present a lower bound of 2 on the competitive ratio of any online algorithm. Motivated by the lower bound, we consider a semi-online variant where upon arrival of a job of size p, we are allowed to migrate jobs of total size at most a constant times p. This constant is called the migration factor of the algorithm. Algorithms with small migration factors are a common approach to bridge the performance of online algorithms and offline algorithms. One can obtain algorithms with a constant migration factor by rounding the size of each incoming job and then applying an ordinal algorithm to the resulting rounded instance. With this in mind, we also consider the framework of ordinal algorithms and characterize the competitive ratio that can be achieved using the aforementioned approaches. More specifically, we show that in both cases, one can get a competitive ratio that is strictly lower than 2, which is the bound from the standard online setting. On the other hand, we prove that no PTAS is possible.

Original language | English |
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Title of host publication | 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022 |

Editors | Petra Berenbrink, Benjamin Monmege |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772228 |

DOIs | |

State | Published - 1 Mar 2022 |

Event | 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022 - Virtual, Marseille, France Duration: 15 May 2022 → 18 May 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 219 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022 |
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Country/Territory | France |

City | Virtual, Marseille |

Period | 15/05/22 → 18/05/22 |

### Bibliographical note

Funding Information:Marten Maack: Partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-The-Fly Computing” under the project number 160364472 – SFB 901/3.

Funding Information:

Funding L. Epstein and A. Levin were partially supported by GIF – the German-Israeli Foundation for Scientific Research and Development (grant number I-1366-407. 6/2016 on “Polynomial Migration for Online Scheduling”). Alexandra Lassota: A. Lassota was supported by the Swiss National Science Foundation within the project Lattice algorithms and Integer Programming (200021_185030/1). Asaf Levin: A. Levin was also partially supported by a grant from ISF – Israeli Science Foundation (grant number 308/18).

Funding Information:

L. Epstein and A. Levin were partially supported by GIF ? the German-Israeli Foundation for Scientific Research and Development (grant number I-1366-407. 6/2016 on ?Polynomial Migration for Online Scheduling?). Alexandra Lassota: A. Lassota was supported by the Swiss National Science Foundation within the project Lattice algorithms and Integer Programming (200021_185030/1). Asaf Levin: A. Levin was also partially supported by a grant from ISF ? Israeli Science Foundation (grant number 308/18).

Publisher Copyright:

© Leah Epstein, Alexandra Lassota, Asaf Levin, Marten Maack, and Lars Rohwedder.

## Keywords

- Cardinality Constrained Scheduling
- Lower Bounds
- Makespan Minimization
- Migration
- Online Algorithms
- Ordinal Algorithms
- Pure Online

## ASJC Scopus subject areas

- Software