Weighted Maximum Independent Set of Geometric Objects in Turnstile Streams

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.