License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.64
URN: urn:nbn:de:0030-drops-126679
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12667/
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Bakshi, Ainesh ; Chepurko, Nadiia ; Woodruff, David P.

Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams

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LIPIcs-APPROX64.pdf (0.7 MB)

Abstract

We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let α be the cardinality of the largest independent set. Our goal is to estimate α in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating β, the largest value of a weighted independent set. We initialize the study of this problem in the turnstile streaming model (insertions and deletions) and provide the first algorithms for estimating α and β.
For unit-length intervals, we obtain a (2+ε)-approximation to α and β in poly(log(n)/ε) space. We also show a matching lower bound. Combined with the 3/2-approximation for insertion-only streams by Cabello and Perez-Lanterno [Cabello and Pérez-Lantero, 2017], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a (8√3/π)-approximation to α and β in poly(log(n)/ε) space, which is closely related to the hexagonal circle packing constant.
Finally, we provide algorithms for estimating α for arbitrary-length intervals under a bounded intersection assumption and study the parameterized space complexity of estimating α and β, where the parameter is the ratio of maximum to minimum interval length.

BibTeX - Entry

@InProceedings{bakshi_et_al:LIPIcs:2020:12667,
  author =	{Ainesh Bakshi and Nadiia Chepurko and David P. Woodruff},
  title =	{{Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{64:1--64:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Jaros{\l}aw Byrka and Raghu Meka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12667},
  URN =		{urn:nbn:de:0030-drops-126679},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.64},
  annote =	{Keywords: Weighted Maximum Independent Set, Geometric Graphs, Turnstile Streams}
}

Keywords: Weighted Maximum Independent Set, Geometric Graphs, Turnstile Streams
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)
Issue Date: 2020
Date of publication: 11.08.2020

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