Better Diameter Algorithms for Bounded VC-dimension Graphs and Geometric Intersection Graphs

TL;DR

This paper introduces improved algorithms for computing the diameter in graphs with bounded VC-dimension, including geometric intersection graphs, achieving faster, simpler solutions with subquadratic time complexity.

Contribution

The authors develop a unified framework that enhances diameter algorithms for bounded VC-dimension graphs and extends to geometric intersection graphs, improving speed and simplicity.

Findings

Achieved subquadratic time complexity for diameter in bounded VC-dimension graphs.

Extended the framework to geometric intersection graphs, including unit squares and convex polygons.

Provided a partial answer to an open problem on diameter algorithms for geometric intersection graphs.

Abstract

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.