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

Abstract

We give the first truly subquadratic time algorithm, with $O^{\ast }(n^{2-1/18})$ 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 $\tilde{O}(m{n}^{1-1/(2d)})$ 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 $\tilde{O}(n^{2-1/12})$ 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.)

FOCS 2025