Online two-dimensional load balancing

Ilan Cohen, Sungjin Im, Debmalya Panigrahi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


In this paper, we consider the problem of assigning 2-dimensional vector jobs to identical machines online so to minimize the maximum load on any dimension of any machine. For arbitrary number of dimensions d, this problem is known as vector scheduling, and recent research has established the optimal competitive ratio as O( logloglogdd ) (Im et al. FOCS 2015, Azar et al. SODA 2018). But, these results do not shed light on the situation for small number of dimensions, particularly for d = 2 which is of practical interest. In this case, a trivial analysis shows that the classic list scheduling greedy algorithm has a competitive ratio of 3. We show the following improvements over this baseline in this paper: We give an improved, and tight, analysis of the list scheduling algorithm establishing a competitive ratio of 8/3 for two dimensions. If the value of opt is known, we improve the competitive ratio to 9/4 using a variant of the classic best fit algorithm for two dimensions. For any fixed number of dimensions, we design an algorithm that is provably the best possible against a fractional optimum solution. This algorithm provides a proof of concept that we can simulate the optimal algorithm online up to the integrality gap of the natural LP relaxation of the problem.

Original languageEnglish
Title of host publication47th International Colloquium on Automata, Languages, and Programming, ICALP 2020
EditorsArtur Czumaj, Anuj Dawar, Emanuela Merelli
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771382
StatePublished - 1 Jun 2020
Externally publishedYes
Event47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 - Virtual, Online, Germany
Duration: 8 Jul 202011 Jul 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference47th International Colloquium on Automata, Languages, and Programming, ICALP 2020
CityVirtual, Online

Bibliographical note

Publisher Copyright:
© Ilan Cohen, Sungjin Im, and Debmalya Panigrahi; licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).


  • Multi-dimensional
  • Online algorithms
  • Scheduling


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