eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-23
51:1
51:18
10.4230/LIPIcs.ESA.2024.51
article
Better Diameter Algorithms for Bounded VC-Dimension Graphs and Geometric Intersection Graphs
Duraj, Lech
1
https://orcid.org/0000-0002-0004-3751
Konieczny, Filip
1
https://orcid.org/0009-0001-9397-0289
Potępa, Krzysztof
1
https://orcid.org/0009-0002-8230-6961
Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Poland
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 𝒪{(k ⋅ n^{1-1/d} ⋅ m ⋅ 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 𝒪{(n^{7/4} ⋅ 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol308-esa2024/LIPIcs.ESA.2024.51/LIPIcs.ESA.2024.51.pdf
Graph Diameter
Geometric Intersection Graphs
Vapnik-Chervonenkis Dimension