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Online Disjoint Set Covers: Randomization Is Not Necessary

Authors Marcin Bienkowski , Jarosław Byrka , Łukasz Jeż

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Author Details

Marcin Bienkowski
  • University of Wrocław, Poland
Jarosław Byrka
  • University of Wrocław, Poland
Łukasz Jeż
  • University of Wrocław, Poland

Funding

Supported by Polish National Science Centre grants 2022/45/B/ST6/00559, 2020/39/B/ST6/01641, and 2020/39/B/ST6/01679.

Acknowledgements

The authors are grateful to the anonymous reviewers for their insightful comments.

Cite As Get BibTex

Marcin Bienkowski, Jarosław Byrka, and Łukasz Jeż. Online Disjoint Set Covers: Randomization Is Not Necessary. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.18

Abstract

In the online disjoint set covers problem, the edges of a hypergraph are revealed online, and the goal is to partition them into a maximum number of disjoint set covers. That is, n nodes of a hypergraph are given at the beginning, and then a sequence of hyperedges (subsets of [n]) is presented to an algorithm. For each hyperedge, an online algorithm must assign a color (an integer). Once an input terminates, the gain of the algorithm is the number of colors that correspond to valid set covers (i.e., the union of hyperedges that have that color contains all n nodes).
We present a deterministic online algorithm that is O(log² n)-competitive, exponentially improving on the previous bound of O(n) and matching the performance of the best randomized algorithm by Emek et al. [ESA 2019].
For color selection, our algorithm uses a novel potential function, which can be seen as an online counterpart of the derandomization method of conditional probabilities and pessimistic estimators. There are only a few cases where derandomization has been successfully used in the field of online algorithms. In contrast to previous approaches, our result extends to the following new challenges: (i) the potential function derandomizes not only the Chernoff bound, but also the coupon collector’s problem, (ii) the value of Opt of the maximization problem is not bounded a priori, and (iii) we do not produce a fractional solution first, but work directly on the input.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Disjoint Set Covers
  • Derandomization
  • pessimistic Estimator
  • potential Function
  • online Algorithms
  • competitive Analysis

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References

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