Real-weighted Diameter and Eccentricity of Minor-free and Bounded VC-dimension Graphs in Truly Subquadratic Time
Abstract
We present the first truly subquadratic time algorithm to compute diameter and eccentricity in real-weighted directed graphs with constant distance VC-dimension and strongly sublinear-sized balanced separators. This runs in O(n2-1/(2h-2)polylog(n)) time for real-weighted Kh-minor-free digraphs. Prior to this work, truly subquadratic time computation of diameter was only known for real-weighted planar graphs, while extensions to broader classes like minor-free graphs were restricted to unweighted settings. In particular, existing algorithms that use VC-dimension [Ducoffe, Habib, Viennot; SICOMP 2022][Le, Wulff-Nilsen; SODA 2024][Chan, Chang, Gao, Le, Kisfaludi-Bak, Zheng; FOCS 2025] work with small integer weights, but do not naturally generalize to real weights. We overcome this barrier by introducing a randomized search-to-decision reduction, demonstrating that VC-dimension is a sufficiently powerful tool in the real-weighted regime.
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