We present an algorithm that computes the girth of the intersection graph of $n$ given line segments in the plane in $O(n^{1.483})$ expected time. This is the first such algorithm with $O(n^{3/2-\varepsilon})$ running time for a positive constant $\varepsilon$, and makes progress towards an open question posed by Chan (SODA 2023). The main techniques include (i)~the usage of recent subcubic algorithms for bounded-difference min-plus matrix multiplication, and (ii)~an interesting variant of the planar graph separator theorem. The result extends to intersection graphs of connected algebraic curves or semialgebraic sets of constant description complexity.

翻译：我们提出一种算法，可在期望时间 $O(n^{1.483})$ 内计算平面内给定 $n$ 条线段相交图的围长。这是首个对于正常数 $\varepsilon$ 具有 $O(n^{3/2-\varepsilon})$ 运行时间的此类算法，推进了 Chan（SODA 2023）提出的未解决问题。主要技术包括：(i) 使用近期有界差 min-plus 矩阵乘法的次立方算法，以及 (ii) 平面图分离定理的一个有趣变体。该结果可推广至连通代数曲线或常数描述复杂度的半代数集的相交图。