2026-07-22T01:57:57Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16029 2024-03-06T10:56:46Z ddc:004 open_access

Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs Bringmann, Karl Kisfaludi‑Bak, Sándor Künnemann, Marvin Nusser, André Parsaeian, Zahra Hardness in P Geometric Intersection Graph Graph Diameter Orthogonal Vectors Hyperclique Detection 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Karl Bringmann and Sándor Kisfaludi‑Bak and Marvin Künnemann and André Nusser and Zahra Parsaeian 2022 Is Part Of LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2022.21 urn:nbn:de:0030-drops-160294 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.21 eng https://creativecommons.org/licenses/by/4.0/legalcode