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Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs

Authors Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, André Nusser , Zahra Parsaeian

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Author Details

Karl Bringmann
  • Universität des Saarlandes, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Sándor Kisfaludi‑Bak
  • Aalto University, Espoo, Finland
Marvin Künnemann
  • Institute for Theoretical Studies, ETH Zürich, Switzerland
André Nusser
  • BARC, University of Copenhagen, Denmark
Zahra Parsaeian
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Bringmann, Karl: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).
  • Kisfaludi‑Bak, Sándor: Part of this research was conducted while the author was at the Max Planck Institute for Informatics, and part of it while he was at the Institute for Theoretical Studies, ETH Zürich.
  • Künnemann, Marvin: Research supported by Dr. Max Rössler, by the Walter Haefner Foundation, and by the ETH Zürich Foundation. Part of this research was conducted while the author was at the Max Planck Institute for Informatics.
  • Nusser, André: Part of this research was conducted while the author was at Saarbrücken Graduate School of Computer Science and Max Planck Institute for Informatics. The author is supported by the VILLUM Foundation grant 16582.

Cite As Get BibTex

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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