# Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams

## TL;DR

This paper introduces the first algorithms for estimating the maximum independent set size of geometric objects like intervals and disks in turnstile streaming models, providing approximation guarantees and space complexity bounds.

## Contribution

It presents novel algorithms for estimating maximum independent sets of geometric objects in turnstile streams, including approximation ratios and space bounds, and explores parameterized complexity.

## Key findings

- Achieves a (2+ε)-approximation for unit-length intervals in poly(log(n)/ε) space.
- Provides a matching lower bound, demonstrating optimality of the approximation.
- Extends results to disks and arbitrary intervals under certain assumptions.

## 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 $\alpha$ be the cardinality of the largest independent set. Our goal is to estimate $\alpha$ 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 $\beta$, 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 $\alpha$ and $\beta$. For unit-length intervals, we obtain a $(2+\epsilon)$-approximation to $\alpha$ and $\beta$ in poly$(\frac{\log(n)}{\epsilon})$ space. We also show a matching lower bound. Combined with the $3/2$-approximation for insertion-only streams by Cabello and Perez-Lanterno [CP15], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a $\left(\frac{8\sqrt{3}}{\pi}\right)$-approximation to $\alpha$ and $\beta$ in poly$(\log(n), \epsilon^{-1})$ space, which is closely related to the hexagonal circle packing constant. We provide algorithms for estimating $\alpha$ for arbitrary-length intervals under a bounded intersection assumption and study the parameterized space complexity of estimating $\alpha$ and $\beta$, where the parameter is the ratio of maximum to minimum interval length.