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DOI: 10.4230/LIPIcs.ESA.2024.58

Random-Order Online Independent Set of Intervals and Hyperrectangles

Authors: Mohit Garg, Debajyoti Kar, and Arindam Khan

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

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.

Cite as

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)

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@InProceedings{garg_et_al:LIPIcs.ESA.2024.58,
  author =	{Garg, Mohit and Kar, Debajyoti and Khan, Arindam},
  title =	{{Random-Order Online Independent Set of Intervals and Hyperrectangles}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{58:1--58:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.58},
  URN =		{urn:nbn:de:0030-drops-211298},
  doi =		{10.4230/LIPIcs.ESA.2024.58},
  annote =	{Keywords: Online Algorithms, Random-Order Model, Maximum Independent Set of Rectangles, Hyperrectangles, Fat Objects, Interval Scheduling}
}

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DOI: 10.4230/LIPIcs.ESA.2022.71

Approximation Algorithms for Round-UFP and Round-SAP

Authors: Debajyoti Kar, Arindam Khan, and Andreas Wiese

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract

We study Round-UFP and Round-SAP, two generalizations of the classical Bin Packing problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of jobs where for each job we are given a demand and a subpath. In Round-UFP, the goal is to find a packing of all jobs into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of jobs on any edge does not exceed the capacity of the respective edge. In Round-SAP, the jobs are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below the capacity profile of the edges. We show that in contrast to Bin Packing, both problems do not admit an asymptotic polynomial-time approximation scheme (APTAS), even when all edge capacities are equal. However, for this setting, we obtain asymptotic (2+ε)-approximations for both problems. For the general case, we obtain an O(log log n)-approximation algorithm and an O(log log 1/δ)-approximation under (1+δ)-resource augmentation for both problems. For the intermediate setting of the no bottleneck assumption (i.e., the maximum job demand is at most the minimum edge capacity), we obtain an absolute 12- and an asymptotic (16+ε)-approximation algorithm for Round-UFP and Round-SAP, respectively.

Cite as

Debajyoti Kar, Arindam Khan, and Andreas Wiese. Approximation Algorithms for Round-UFP and Round-SAP. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 71:1-71:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kar_et_al:LIPIcs.ESA.2022.71,
  author =	{Kar, Debajyoti and Khan, Arindam and Wiese, Andreas},
  title =	{{Approximation Algorithms for Round-UFP and Round-SAP}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{71:1--71:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.71},
  URN =		{urn:nbn:de:0030-drops-170098},
  doi =		{10.4230/LIPIcs.ESA.2022.71},
  annote =	{Keywords: Approximation Algorithms, Scheduling, Rectangle Packing}
}

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Documents authored by Kar, Debajyoti

Random-Order Online Independent Set of Intervals and Hyperrectangles

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Approximation Algorithms for Round-UFP and Round-SAP

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