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# Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs

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## Cite As

Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, André Nusser, and Zahra Parsaeian. Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.21

## Abstract

We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in d-dimensional Euclidean space, such as balls, segments, or hypercubes, and whose edges correspond to pairs of intersecting shapes. The diameter of a graph is the largest distance realized by a pair of vertices in the graph. Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since in the related class of planar graphs the diameter can be computed in 𝒪̃(n^{5/3}) time [Cabello 2019, Gawrychowski et al. 2021]. In this work we (conditionally) rule out sub-quadratic algorithms in several classes of intersection graphs, i.e., algorithms of running time 𝒪(n^{2-δ}) for some δ > 0. In particular, there are no sub-quadratic algorithms already for fat objects in small dimensions: unit balls in ℝ³ or congruent equilateral triangles in ℝ². For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic approximations already in ℝ². It seems that the hardness of approximation may also depend on dimensionality: for axis-parallel unit hypercubes in ℝ^{12}, distinguishing between diameter 2 and 3 needs quadratic time (ruling out (3/2-ε)- approximations), whereas for axis-parallel unit squares, we give an algorithm that distinguishes between diameter 2 and 3 in near-linear time. Note that many of our lower bounds match the best known algorithms up to sub-polynomial factors. Ultimately, this fine-grained perspective may enable us to determine for which shapes we can have efficient algorithms and approximation schemes for diameter computation.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Hardness in P
• Geometric Intersection Graph
• Graph Diameter
• Orthogonal Vectors
• Hyperclique Detection

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## References

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