Real-weighted Diameter and Eccentricity of Minor-free and Bounded VC-dimension Graphs in Truly Subquadratic Time

2026-07-02

AI Summary

This work addresses the problem of efficiently computing the diameter and eccentricities in real-weighted directed graphs. It presents the first truly subquadratic-time algorithms for graph classes admitting constant-distance VC dimension and strongly sublinear balanced separators—such as $K_h$-minor-free graphs. By extending VC dimension–based techniques to the real-weighted setting and integrating randomized search-to-decision reductions with graph decomposition and balanced separator methods, the algorithm achieves a running time of $O(n^{2-1/(2h-2)} \text{polylog}\,n)$ on $K_h$-minor-free directed graphs. This result overcomes the prior limitation to unweighted or integer-weighted graphs, significantly broadening the scope of graph classes amenable to efficient diameter and eccentricity computation.

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 $O(n^{2-1/(2h-2)}\textrm{polylog}(n))$ time for real-weighted $K_h$-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.

Problem

• diameter
• eccentricity
• real-weighted graphs
• minor-free graphs
• VC-dimension

Innovation

• real-weighted graphs
• truly subquadratic time
• VC-dimension
• minor-free graphs
• randomized search-to-decision reduction