A fundamental question in computational geometry is for a dynamic collection of geometric objects in Euclidean space, whether it is possible to maintain a maximum independent set in polylogarithmic update time. Already, for a set of intervals, it is known that no dynamic algorithm can maintain an exact maximum independent set with 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 update time. First, we show that for a fully dynamic set of $n$ unit disks in the plane, a $12$-approximate maximum independent set can be maintained with worst-case update time $O(\log^2 n)$, and optimal output-sensitive reporting. Moreover, 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. Our main result is that for a fully dynamic set of disks of arbitrary radii in the plane, an $O(1)$-approximate maximum independent set can be maintained in polylogarithmic expected amortized update time.

翻译：计算几何中的一个基本问题是：对于欧氏空间中几何对象的动态集合，是否可能以多对数更新时间维护一个最大独立集。已知对于一组区间，不存在任何动态算法能以亚线性更新时间保持精确的最大独立集。因此，典型目标是在更新时间和解规模之间探索权衡。近年来，针对各类几何对象（如区间、超立方体、超矩形和胖对象）理解该问题已投入大量努力。我们首次提出平面中任意半径圆盘的完全动态近似算法，能以多对数更新时间维护常数因子近似最大独立集。首先，我们证明对于平面中$n$个单位圆盘的完全动态集合，可以以最坏情况更新时间$O(\log^2 n)$和最优输出敏感报告方式维护一个$12$-近似最大独立集。进一步地，该结果可推广至任意固定维度$d$中尺寸可比的胖对象，其中近似比取决于维度和胖度参数。我们的主要结果是：对于平面中任意半径圆盘的完全动态集合，能以多对数期望分摊更新时间维护一个$O(1)$-近似最大独立集。