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Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams

Authors Ainesh Bakshi, Nadiia Chepurko, David P. Woodruff

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Author Details

Ainesh Bakshi
  • Carnegie Mellon University, Pittsburgh, PA, USA
Nadiia Chepurko
  • MIT, Cambridge, MA, USA
David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

Ainesh Bakshi and David Woodruff acknowledge support in part from NSF No. CCF-1815840.

Acknowledgements

Part of this work was done while Ainesh Bakshi and David Woodruff were visiting the Simons Institute for the Theorem of Computing.

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Ainesh Bakshi, Nadiia Chepurko, and David P. Woodruff. Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 64:1-64:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.64

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Streaming models
Keywords
  • Weighted Maximum Independent Set
  • Geometric Graphs
  • Turnstile Streams

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