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

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 \(|E(G)|\ge \alpha \, |V(G)|^2\) for some fixed \(\alpha \gt 0\) . 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 \(\varepsilon \gt 0\) , returns an integer g such that G has an embedding in a surface of genus g , and this is ɛ-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 \(O(f_1(\varepsilon)\,n^2)\) . Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [ 25 ]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [ 17 ] and to Fox, Lovász, and Zhao [ 14 , 15 ]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ɛ-approximation \(\nu (G)\) for the maximum number of edge-disjoint triangles in G . The value \(\nu (G)\) can be computed by solving a linear program whose size is bounded by certain value \(f_2(\varepsilon)\) depending only on ɛ. After solving the linear program, the genus can be approximated (see Corollary 1.7 ). The proof of this result is long and will be of independent interest in topological graph theory.