Beyond the Euler characteristic: Approximating the genus of general graphs

TL;DR

This paper introduces the first non-trivial polynomial-time approximation algorithm for computing the Euler genus of graphs, improving over trivial bounds and enabling algorithms on graphs with unknown embeddings.

Contribution

It presents a polynomial-time algorithm that approximates the Euler genus within a polynomial factor, advancing the understanding of graph genus approximation.

Findings

First non-trivial approximation algorithm for Euler genus

Implications for algorithms on graphs with unknown embeddings

Achieves a polynomial factor approximation

Abstract

Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by [Thomassen '89] and a linear-time fixed-parameter algorithm has been obtained by [Mohar '99]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of an $O(1)$-approximation is not ruled out, the currently best-known upper bound is a trivial $O(n/g)$-approximation that follows from bounds on the Euler characteristic. In this paper, we give the first non-trivial approximation algorithm for this problem. Specifically, we present a polynomial-time algorithm which given a graph $G$ of Euler genus $g$ outputs an embedding of $G$ into a surface of Euler genus $g^{O(1)}$. Combined with the above $O(n/g)$-approximation, our result also implies a $O(n^{1-\alpha})$-approximation, for some universal constant $\alpha>0$. Our approximation algorithm also has implications for the design of algorithms on graphs of small genus. Several of these algorithms require that an embedding of the graph into a surface of small genus is given as part of the input. Our result implies that many of these algorithms can be implemented even when the embedding of the input graph is unknown.