We develop a framework for algorithms finding the diameter in graphs of bounded distance Vapnik-Chervonenkis dimension, in (parameterized) subquadratic 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. [SODA'20, SIGCOMP'22], improving their technique. With our approach the algorithms become simpler and faster, working in $\mathcal{O}(k \cdot n^{1-1/d} \cdot m \cdot \mathrm{polylog}(n))$ time complexity for the graph on $n$ vertices and $m$ edges, where $k$ is the diameter and $d$ is the distance VC-dimension of the graph. Furthermore, it allows us to use the improved 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 partially answer a question posed by Bringmann et al. [SoCG'22], finding an $\mathcal{O}(n^{7/4} \cdot \mathrm{polylog}(n))$ parameterized diameter algorithm for unit square intersection graph of size $n$, as well as a more general algorithm for convex polygon intersection graphs.

翻译：我们开发了一个框架，用于在有界距离Vapnik-Chervonenkis维度的图中以（参数化）次二次时间复杂度求解直径问题。有界距离VC维图类范围广泛，包含例如所有免子图。我们在Ducoffe等人[SODA'20, SIGCOMP'22]的工作基础上改进其技术。通过我们的方法，算法变得更简洁高效，对于具有$n$个顶点和$m$条边的图，时间复杂度为$\mathcal{O}(k \cdot n^{1-1/d} \cdot m \cdot \mathrm{polylog}(n))$，其中$k$为直径，$d$为图的距离VC维度。此外，该框架使我们能在更一般的场景中应用改进后的技术。特别地，我们将此框架应用于几何相交图，即顶点对应平面上相同几何对象且邻接关系由相交定义的图。通过在此类图中应用我们的方法，我们部分回答了Bringmann等人[SoCG'22]提出的问题：针对规模为$n$的单位正方形相交图，给出了$\mathcal{O}(n^{7/4} \cdot \mathrm{polylog}(n))$参数化直径算法，并为凸多边形相交图提供了更通用的算法。