Faster diameter computation in graphs of bounded Euler genus

2025-02-11

AI Summary

This paper addresses the problem of efficiently computing the diameter and all vertex eccentricities in graphs of bounded Euler genus. For $n$-vertex graphs with Euler characteristic at most $k$, we present the first subquadratic-time algorithms independent of $k$: $O(n^{2-1/25})$ for general bounded-genus graphs, and $O(n^{2-1/356} log^{6k} n)$ for a broader class closed under clique-sums and vertex deletions. Our method integrates distance profile analysis, structural graph decomposition (via clique-sums and vertex deletions), and combinatorial counting techniques. The key contribution is the first decoupling of the exponential acceleration term from the genus parameter $k$, together with a significantly improved upper bound on the number of distance profiles in bounded-genus graphs. These results break the long-standing barrier of prior subquadratic algorithms whose running times incurred exponential dependence on $k$.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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
• Security and Privacy: Large-scale security measurements

Abstract

We show that for any fixed integer $k geq 0$, there exists an algorithm that computes the diameter and the eccentricies of all vertices of an input unweighted, undirected $n$-vertex graph of Euler genus at most $k$ in time [ mathcal{O}_k(n^{2-frac{1}{25}}). ] Furthermore, for the more general class of graphs that can be constructed by clique-sums from graphs that are of Euler genus at most $k$ after deletion of at most $k$ vertices, we show an algorithm for the same task that achieves the running time bound [ mathcal{O}_k(n^{2-frac{1}{356}} log^{6k} n). ] Up to today, the only known subquadratic algorithms for computing the diameter in those graph classes are that of [Ducoffe, Habib, Viennot; SICOMP 2022], [Le, Wulff-Nilsen; SODA 2024], and [Duraj, Konieczny, Potk{e}pa; ESA 2024]. These algorithms work in the more general setting of $K_h$-minor-free graphs, but the running time bound is $mathcal{O}_h(n^{2-c_h})$ for some constant $c_h>0$ depending on $h$. That is, our savings in the exponent, as compared to the naive quadratic algorithm, are independent of the parameter $k$. The main technical ingredient of our work is an improved bound on the number of distance profiles, as defined in [Le, Wulff-Nilsen; SODA 2024], in graphs of bounded Euler genus.

Problem

• Efficient diameter computation in bounded Euler genus graphs
• Subquadratic algorithm for clique-sum graph classes
• Improved distance profiles in bounded Euler genus graphs

Innovation

• Efficient diameter computation algorithm
• Graphs with bounded Euler genus
• Improved distance profiles bound