Charting the Diameter Computation Landscape of Geometric Intersection Graphs in Three Dimensions and Higher

2026-03-23

AI Summary

This work addresses the long-standing open problem of whether truly subquadratic algorithms exist for computing the diameter in intersection graphs of three- or higher-dimensional geometric objects, such as unit balls and unit cubes. Focusing on small-diameter cases, the paper leverages the geometric structure of pseudoline arrangements to design efficient algorithms and establishes conditional lower bounds under the Orthogonal Vectors (OV) hypothesis. The main contributions include the first truly subquadratic algorithm for Diameter-3 in 3D unit cube intersection graphs and a near-linear algorithm for Diameter-2. Conversely, it proves that Diameter-3 in 3D unit ball intersection graphs admits no truly subquadratic algorithm under the OV conjecture. These results are further generalized to axis-aligned boxes in arbitrary dimensions, yielding a unified framework for both algorithms and conditional lower bounds for Diameter-2 and Diameter-3.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [SoCG '22] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter $Δ$ is at least $Ω(\log n)$, hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: - A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. - A truly subquadratic time lower bound for \Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be truly subquadratic hard when the diameter is $Ω(\log n)$. - A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D. - A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

Problem

• diameter computation
• geometric intersection graphs
• higher dimensions
• unit cubes
• unit balls

Innovation

• geometric intersection graphs
• diameter computation
• subquadratic algorithms
• unit cubes
• Orthogonal Vector hypothesis