eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-11
64:1
64:22
10.4230/LIPIcs.APPROX/RANDOM.2020.64
article
Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams
Bakshi, Ainesh
1
Chepurko, Nadiia
2
Woodruff, David P.
1
Carnegie Mellon University, Pittsburgh, PA, USA
MIT, Cambridge, MA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol176-approx-random2020/LIPIcs.APPROX-RANDOM.2020.64/LIPIcs.APPROX-RANDOM.2020.64.pdf
Weighted Maximum Independent Set
Geometric Graphs
Turnstile Streams