Let $G$ be an intersection graph of $n$ geometric objects in the plane. We show that a maximum matching in $G$ can be found in $O(\rho^{3\omega/2}n^{\omega/2})$ time with high probability, where $\rho$ is the density of the geometric objects and $\omega>2$ is a constant such that $n \times n$ matrices can be multiplied in $O(n^\omega)$ time. The same result holds for any subgraph of $G$, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in $O(n^{\omega/2})$ time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in $[1, \Psi]$ can be found in $O(\Psi^6\log^{11} n + \Psi^{12 \omega} n^{\omega/2})$ time with high probability.

翻译：设$G$为平面内$n$个几何对象的交图。我们证明，在高概率下，可在$O(\rho^{3\omega/2}n^{\omega/2})$时间内找到$G$中的最大匹配，其中$\rho$为几何对象的密度，$\omega>2$为常数，满足$n \times n$矩阵可在$O(n^\omega)$时间内完成乘法运算。该结论对$G$的任意子图同样成立，仅需保证几何表示可用。为此，我们融合了代数方法（即通过高斯消元计算矩阵秩）与几何交图具有小分隔符的特性。进一步表明，在许多重要情形下，一般几何交图中的最大匹配问题可归约为有界密度情形。特别地，高概率下，平面内任意凸对象平移族交图中的最大匹配可在$O(n^{\omega/2})$时间内求得；而半径在$[1, \Psi]$范围内的平面圆盘族交图中的最大匹配，高概率下可在$O(\Psi^6\log^{11} n + \Psi^{12 \omega} n^{\omega/2})$时间内求得。