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.

翻译：一个基本问题是：对于欧几里得空间中动态几何对象集合，能否在对数级更新时间内维护最大独立集。已知即使对于区间集合，任何动态算法都无法在次线性更新时间内维护精确最大独立集。因此，典型目标是探索更新时间与解规模之间的权衡。近年来，已有大量研究致力于理解各类几何对象（如区间、超立方体、超矩形和胖对象）的这一核心问题。本文提出了首个完全动态近似算法，适用于平面上任意半径的圆盘，能够在对数级期望摊销更新时间内保持常数因子近似最大独立集。此外，对于平面上单位半径圆盘的完全动态集合（规模为$n$），我们证明了可在最坏情况更新时间$O(\log n)$内维护一个12-近似的最大独立集，并支持最优输出敏感报告。该结果可推广至任意固定维度$d$中尺寸相当的胖对象，其近似比取决于维度和胖度参数。此外，我们指出：即使对于平面上单位半径圆盘的动态集合，在标准复杂度假设下，无法在真正次线性更新时间内维护$O(1+\varepsilon)$-近似最大独立集。