Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

TL;DR

This paper introduces the first fully dynamic approximation algorithms for maintaining near-optimal independent sets of disks in the plane with polylogarithmic update times, advancing the understanding of dynamic geometric graph algorithms.

Contribution

It presents the first fully dynamic approximation algorithm for disks of arbitrary radii with polylogarithmic update time and an improved worst-case update time for unit disks, extending to fat objects in fixed dimensions.

Findings

Maintains constant-factor approximate maximum independent sets in polylogarithmic expected amortized time.

Achieves worst-case $O( ext{log} n)$ update time for unit disks with optimal reporting.

Proves the impossibility of maintaining near-constant approximation in truly sublinear time under standard assumptions.

Abstract

A fundamental question is whether one can maintain a maximum independent set in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set in sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects. We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate maximum independent set in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of $n$ disks of unit radius in the plane, we show that a $12$-approximate maximum independent set can be maintained with worst-case update time $O(\log n)$, and optimal output-sensitive reporting. This result generalizes to fat objects of comparable sizes in any fixed dimension $d$, where the approximation ratio depends on the dimension and the fatness parameter. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain $O(1+\varepsilon)$-approximate maximum independent set in truly sublinear update time, under standard complexity assumptions.