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Random-Order Online Independent Set of Intervals and Hyperrectangles

Authors Mohit Garg, Debajyoti Kar , Arindam Khan

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Author Details

Mohit Garg
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India
Debajyoti Kar
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India
Arindam Khan
  • Department of Computer Science and Automation, Indian Institute of Science, Bengaluru, India

Funding

  • Garg, Mohit: Supported by a fellowship from the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).
  • Kar, Debajyoti: Supported by a Google PhD Fellowship.
  • Khan, Arindam: Supported in part by Google India Research Award, SERB Core Research Grant (CRG/2022/001176) on "Optimization under Intractability and Uncertainty", and the Walmart Center for Tech Excellence at IISc (CSR Grant WMGT-23-0001).

Acknowledgements

The authors wish to thank Jaikumar Radhakrishnan for generously sharing his insights on hypergeometric concentration bounds, Rahul Saladi for helping in constructing fast and elegant data structures, and K.V.N. Sreenivas for some helpful discussions.

Cite AsGet BibTex

Mohit Garg, Debajyoti Kar, and Arindam Khan. Random-Order Online Independent Set of Intervals and Hyperrectangles. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 58:1-58:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.58

Abstract

In the Maximum Independent Set of Hyperrectangles problem, we are given a set of n (possibly overlapping) d-dimensional axis-aligned hyperrectangles, and the goal is to find a subset of non-overlapping hyperrectangles of maximum cardinality. For d = 1, this corresponds to the classical Interval Scheduling problem, where a simple greedy algorithm returns an optimal solution. In the offline setting, for d-dimensional hyperrectangles, polynomial time (log n)^{O(d)}-approximation algorithms are known [Chalermsook and Chuzhoy, 2009]. However, the problem becomes notably challenging in the online setting, where the input objects (hyperrectangles) appear one by one in an adversarial order, and on the arrival of an object, the algorithm needs to make an immediate and irrevocable decision whether or not to select the object while maintaining the feasibility. Even for interval scheduling, an Ω(n) lower bound is known on the competitive ratio. To circumvent these negative results, in this work, we study the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, where the adversary specifies the set of input objects which then arrive in a uniformly random order. Starting from the prototypical secretary problem, the random-order model has received significant attention to study algorithms beyond the worst-case competitive analysis (see the survey by Gupta and Singla [Anupam Gupta and Sahil Singla, 2020]). Surprisingly, we show that the problem in the random-order model almost matches the best-known offline approximation guarantees, up to polylogarithmic factors. In particular, we give a simple (log n)^{O(d)}-competitive algorithm for d-dimensional hyperrectangles in this model, which runs in O_d̃(n) time. Our approach also yields (log n)^{O(d)}-competitive algorithms in the random-order model for more general objects such as d-dimensional fat objects and ellipsoids. Furthermore, all our competitiveness guarantees hold with high probability, and not just in expectation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Computational geometry
  • Theory of computation → Scheduling algorithms
Keywords
  • Online Algorithms
  • Random-Order Model
  • Maximum Independent Set of Rectangles
  • Hyperrectangles
  • Fat Objects
  • Interval Scheduling

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