Random-Order Online Independent Set of Intervals and Hyperrectangles

Garg, M and Kar, D and Khan, A (2024) Random-Order Online Independent Set of Intervals and Hyperrectangles. In: Leibniz International Proceedings in Informatics, LIPIcs -32nd Annual European Symposium on Algorithms, ESA 2024, 2 September 2024- 4 September 2024, London.

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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 16. 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 40). 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 �d(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. © Mohit Garg, Debajyoti Kar, and Arindam Khan; licensed under Creative Commons License CC-BY 4.0.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to the publisher.
Keywords:	Digital arithmetic; Fractals; Polynomial approximation; Random processes; Scheduling algorithms; Set theory, Competitive algorithms; Fat objects; Hyperrectangles; Interval scheduling; Maximum independent set of rectangle; Maximum independent sets; Offline; On-line algorithms; Random-order model; Simple++, Approximation algorithms
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	24 Oct 2024 12:00
Last Modified:	24 Oct 2024 12:00
URI:	http://eprints.iisc.ac.in/id/eprint/86604

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