On Streaming Algorithms for Geometric Independent Set and Clique

TL;DR

This paper investigates streaming algorithms for geometric independent set and clique problems, establishing hardness results for constant-factor approximations and providing a 4-approximation for axis-aligned rectangles with efficient memory use.

Contribution

It proves the non-existence of constant-factor streaming algorithms for maximum geometric independent set and clique problems under certain conditions, and presents a memory-efficient 4-approximation for rectangles.

Findings

No constant factor approximation for segments or 2-intervals without linear memory.

Existence of a 2-approximation for interval independent sets with logarithmic memory.

A 4-approximation algorithm for maximum independent set of axis-aligned rectangles.

Abstract

We study the maximum geometric independent set and clique problems in the streaming model. Given a collection of geometric objects arriving in an insertion only stream, the aim is to find a subset such that all objects in the subset are pairwise disjoint or intersect respectively. We show that no constant factor approximation algorithm exists to find a maximum set of independent segments or $2$-intervals without using a linear number of bits. Interestingly, our proof only requires a set of segments whose intersection graph is also an interval graph. This reveals an interesting discrepancy between segments and intervals as there does exist a $2$-approximation for finding an independent set of intervals that uses only $O(\alpha(\mathcal{I})\log |\mathcal{I}|)$ bits of memory for a set of intervals $\mathcal{I}$ with $\alpha(\mathcal{I})$ being the size of the largest independent set of $\mathcal{I}$. On the flipside we show that for the geometric clique problem there is no constant-factor approximation algorithm using less than a linear number of bits even for unit intervals. On the positive side we show that the maximum geometric independent set in a set of axis-aligned unit-height rectangles can be $4$-approximated using only $O(\alpha(\mathcal{R})\log |\mathcal{R}|)$ bits.