Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in 2D, the problem is solvable in truly subquadratic time, while for other objects, including unit segments and equilateral triangles in 2D or unit balls and axis-parallel unit cubes in 3D, there is no truly subquadratic time algorithm under the Orthogonal Vector (OV) hypothesis. We undertake a comprehensive study of computing the diameter of geometric intersection graphs for various types of objects. We discover many new irregularities, showing that the landscape is extremely nuanced: the source of hardness is a combination of the object type, the true diameter value, and how the objects intersect with each other. Our highlighted results for the 2D case include: 1. The diameter of non-degenerate, axis-aligned line segments can be computed in truly subquadratic time. Previous hardness result for line segments applies only to degenerate instances. On the other hand, for the degenerate case, we show that a truly subquadratic time algorithm exists when the true diameter is constant. 2. An almost-linear-time algorithm for unit-square graphs of constant diameter. Previous algorithms rely on succinct representation assuming bounded VC-dimension; for such a strategy $Ω(n^{7/4})$ time is an inherent barrier. 3. An $\tilde{O}(n^{4/3})$-time algorithm to decide if the diameter of a unit-disk graph is at most 2. This improves upon the recent algorithm with running time $\tilde{O}(n^{2-1/9})$. 4. Deciding if the diameter of intersection graphs of fat triangles or line segments is at most 2 is truly subquadratic-hard under fine-grained complexity assumptions. Previous lower bounds only hold when deciding if diameter is at most 3.

翻译：计算物体相交图直径是计算几何中的一个基本问题。先前研究表明，直径计算复杂性主要取决于物体类型：对于二维中的单位圆盘和正方形，该问题可在真正次二次时间内解决；而对于其他物体，包括二维中的单位线段和等边三角形，或三维中的单位球体和轴平行单位立方体，在正交向量（OV）假设下不存在真正次二次时间算法。我们对各类物体的几何相交图直径计算进行了全面研究，发现了许多新的不规则性，表明这一领域极为微妙：困难根源是物体类型、真实直径值以及物体相互交叠方式的综合作用。我们强调的二维情形结果包括：1. 非退化轴对齐线段的直径可在真正次二次时间内计算。此前线段的困难结果仅适用于退化实例。另一方面，对于退化情形，我们证明当真实直径为常数时，存在真正次二次时间算法。2. 常数直径单位正方形图的近似线性时间算法。先前算法依赖有界VC维的简洁表示；此类策略固有障碍是$Ω(n^{7/4})$时间。3. 用于判定单位圆盘图直径是否至多为2的$\tilde{O}(n^{4/3})$时间算法。这改进了近期运行时间为$\tilde{O}(n^{2-1/9})$的算法。4. 在细粒度复杂度假设下，判定胖三角形或线段相交图直径是否至多为2是真正次二次困难的。此前下界仅适用于判定直径是否至多为3的情形。