We develop a framework for algorithms finding diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parametrized) sub-quadratic time complexity. The class of bounded distance VC-dimension graphs is wide, including, e.g. all minor-free graphs. We build on the work of Ducoffe et al., improving their technique. With our approach the algorithms become simpler and faster, working in $\widetilde{\mathcal{O}}(k \cdot V^{1-1/d} \cdot E)$ time complexity, where $k$ is the diameter, $d$ is the VC-dimension. Furthermore, it allows us to use the technique in more general setting. In particular, we use this framework for geometric intersection graphs, i.e. graphs where vertices are identical geometric objects on a plane and the adjacency is defined by intersection. Applying our approach for these graphs, we answer a question posed by Bringmann et al., finding a $\widetilde{\mathcal{O}}(n^{7/4})$ parametrized diameter algorithm for unit square intersection graph of size $n$, as well as a more general algorithm for convex polygon intersection graphs.

翻译：我们提出了一个用于有界距离Vapnik-Chervonenkis维图中求解直径问题的算法框架，其时间复杂度为（参数化）次二次复杂度。有界距离VC维图类涵盖广泛，包括所有无环图。我们基于Ducoffe等人的工作，对其技术进行了改进。采用我们的方法后，算法更简洁高效，时间复杂度为$\widetilde{\mathcal{O}}(k \cdot V^{1-1/d} \cdot E)$，其中$k$为直径，$d$为VC维。此外，该框架可应用于更一般的情形。特别地，我们将此框架应用于几何交图，即顶点为平面上相同几何对象、邻接关系由相交定义的图。针对这类图，我们回答了Bringmann等人提出的问题，给出了大小为$n$的单位正方形交图的$\widetilde{\mathcal{O}}(n^{7/4})$参数化直径算法，以及凸多边形交图的更通用算法。