Truly Subquadratic Time Algorithms for Diameter and Related Problems in Graphs of Bounded VC-dimension

2025-10-18

AI Summary

This paper addresses classical distance problems—including diameter computation, vertex eccentricities, Wiener index, and exact distance prediction—on graphs of bounded VC-dimension, such as unit-disk graphs, axis-aligned square intersection graphs, and sparse unweighted graphs. We propose the first truly subquadratic-time generic algorithmic framework for these problems, breaking the long-standing paradigm reliant on sublinear separators. Our approach innovatively integrates low-diameter decompositions, combinatorial structures inherent to bounded-VC-dimension set systems, and geometric data structures to establish a scalable, divide-and-conquer design paradigm. The framework achieves $O^*(n^{2-1/18})$ time for unit-disk graphs, $ ilde{O}(m n^{1-1/(2d)})$ for sparse graphs with average degree $d$, and $ ilde{O}(n^{2-1/12})$ for axis-aligned square intersection graphs. Crucially, this is the first work to deliver truly subquadratic algorithms for all aforementioned problems on these graph classes.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We give the first truly subquadratic time algorithm, with $O^*(n^{2-1/18})$ running time, for computing the diameter of an $n$-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance of a general framework, applicable to different graph families and distance problems. Surprisingly, our framework completely bypasses sublinear separators (or $r$-divisions) which were used in all previous algorithms. Instead, we use low-diameter decompositions in their most elementary form. We also exploit bounded VC-dimension of set systems associated with the input graph, as well as new ideas on geometric data structures. Among the numerous applications of the general framework, we obtain: 1. An $ ilde{O}(mn^{1-1/(2d)})$ time algorithm for computing the diameter of $m$-edge sparse unweighted graphs with constant VC-dimension $d$. The previously known algorithms by Ducoffe, Habib, and Viennot [SODA 2019] and Duraj, Konieczny, and Potȩpa [ESA 2024] are truly subquadratic only when the diameter is a small polynomial. Our result thus generalizes truly subquadratic time algorithms known for planar and minor-free graphs (in fact, it slightly improves the previous time bound for minor-free graphs). 2. An $ ilde{O}(n^{2-1/12})$ time algorithm for computing the diameter of intersection graphs of axis-aligned squares with arbitrary size. The best-known algorithm by Duraj, Konieczny, and Potȩpa [ESA 2024] only works for unit squares and is only truly subquadratic in the low-diameter regime. 3. The first algorithms with truly subquadratic complexity for other distance-related problems, including all-vertex eccentricities, Wiener index, and exact distance oracles. (... truncated to meet the arXiv abstract requirement.)

Problem

• Computing graph diameter efficiently for bounded VC-dimension graphs
• Developing subquadratic algorithms for distance problems in geometric graphs
• Creating fast frameworks for eccentricity and distance oracle computations

Innovation

• Bypasses sublinear separators using low-diameter decompositions
• Leverages bounded VC-dimension of graph set systems
• Introduces novel geometric data structure techniques