Cardinality Constrained Scheduling in Online Models

Authors Leah Epstein, Alexandra Lassota , Asaf Levin, Marten Maack , Lars Rohwedder

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Leah Epstein
  • Department of Mathematics, University of Haifa, Israel
Alexandra Lassota
  • Chair of Discrete Optimization, EPFL, Lausanne, Switzerland
Asaf Levin
  • Faculty of Industrial Engineering and Management, Technion, Haifa, Israel
Marten Maack
  • Heinz Nixdorf Institute & Department of Computer Science, Paderborn University, Germany
Lars Rohwedder
  • School of Business and Economics, Maastricht University, The Netherlands

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Leah Epstein, Alexandra Lassota, Asaf Levin, Marten Maack, and Lars Rohwedder. Cardinality Constrained Scheduling in Online Models. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
  • Cardinality Constrained Scheduling
  • Makespan Minimization
  • Online Algorithms
  • Lower Bounds
  • Pure Online
  • Migration
  • Ordinal Algorithms


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