A \emph{disk graph} is the intersection graph of (closed) disks in the plane. We consider the classic problem of finding a maximum clique in a disk graph. For general disk graphs, the complexity of this problem is still open, but for unit disk graphs, it is well known to be in P. The currently fastest algorithm runs in time $O(n^{7/3+ o(1)})$, where $n$ denotes the number of disks~\cite{EspenantKM23, keil_et_al:LIPIcs.SoCG.2025.63}. Moreover, for the case of disk graphs with $t$ distinct radii, the problem has also recently been shown to be in XP. More specifically, it is solvable in time $O^*(n^{2t})$~\cite{keil_et_al:LIPIcs.SoCG.2025.63}. In this paper, we present algorithms with improved running times by allowing for approximate solutions and by using randomization: - for unit disk graphs, we give an algorithm that, with constant success probability, computes a $(1-\varepsilon)$-approximate maximum clique in expected time $\tilde{O}(n/\varepsilon^2)$; and - for disk graphs with $t$ distinct radii, we give a parameterized approximation scheme that, with a constant success probability, computes a $(1-\varepsilon)$-approximate maximum clique in expected time $\tilde{O}(f(t)\cdot (1/\varepsilon)^{O(t)} \cdot n)$, for some (exponential) function $f(t)$.

翻译：\emph{磁盘图}是平面上（闭）圆盘的交图。我们考虑在磁盘图中寻找最大团这一经典问题。对于一般磁盘图，该问题的计算复杂度仍未解决，但对于单位圆盘图，众所周知其属于P类。目前最快的算法运行时间为 $O(n^{7/3+ o(1)})$，其中 $n$ 表示圆盘数量~\cite{EspenantKM23, keil_et_al:LIPIcs.SoCG.2025.63}。此外，对于具有 $t$ 种不同半径的磁盘图，该问题最近也被证明属于XP类。更具体地说，它可以在 $O^*(n^{2t})$ 时间内求解~\cite{keil_et_al:LIPIcs.SoCG.2025.63}。在本文中，我们通过允许近似解和使用随机化，提出了具有改进运行时间的算法：- 对于单位圆盘图，我们给出一种算法，以恒定成功概率在期望时间 $\tilde{O}(n/\varepsilon^2)$ 内计算 $(1-\varepsilon)$-近似最大团；- 对于具有 $t$ 种不同半径的磁盘图，我们给出一种参数化近似方案，以恒定成功概率在期望时间 $\tilde{O}(f(t)\cdot (1/\varepsilon)^{O(t)} \cdot n)$ 内计算 $(1-\varepsilon)$-近似最大团，其中 $f(t)$ 为某个（指数）函数。