Efficient polynomial-time approximation scheme for the genus of dense graphs

TL;DR

The results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs and extend the algorithm to output an embedding, whose genus is arbitrarily close to the minimum genus.

Abstract

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that $|E(G)|\ge α|V(G)|^2$ for some fixed $α>0$. While a constant factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph $G$ of order $n$ and given $\varepsilon>0$, returns an integer $g$ such that $G$ has an embedding into a surface of genus $g$, and this is $\varepsilon$-close to a minimum genus embedding in the sense that the minimum genus $\mathsf{g}(G)$ of $G$ satisfies: $\mathsf{g}(G)\le g\le (1+\varepsilon)\mathsf{g}(G)$. The running time of the algorithm is $O(f(\varepsilon)\,n^2)$, where $f(\cdot)$ is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is $g$. This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time $O(f_1(\varepsilon)\,n^2)$.

Figures (1)

• Figure 1: A $3$-blossom and a $4$-blossom of length $6$. The cycle $C_1$ is shown by bold edges.

Theorems & Definitions (54)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 2.1: Euler's Formula
• Lemma 2.2
• proof
• Theorem 2.3