Faster diameter computation in graphs of bounded Euler genus

TL;DR

This paper presents faster algorithms for computing the diameter and eccentricities in graphs of bounded Euler genus, improving the exponent in the running time independently of the genus parameter.

Contribution

It introduces subquadratic algorithms for diameter computation in graphs of bounded Euler genus and clique-sums, with savings in the exponent independent of the genus parameter.

Findings

Achieves _k(n^{2-rac{1}{25}}) time for Euler genus at most k.

Extends to clique-sums with _k(n^{2-rac{1}{356}} \u221a^{6k}} n) time.

Main technical advance is an improved bound on the number of distance profiles in such graphs.

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, Pot\k{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.