Better Diameter Algorithms for Bounded VC-dimension Graphs and Geometric Intersection Graphs
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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 𝒪(k · V^1-1/d· 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 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.
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