Efficient polynomial-time approximation scheme for the genus of dense graphs
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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)|≥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 ε>0, returns an integer g such that G has an embedding into a surface of genus g, and this is ε-close to a minimum genus embedding in the sense that the minimum genus 𝗀(G) of G satisfies: 𝗀(G)≤ g≤ (1+ε)𝗀(G). The running time of the algorithm is O(f(ε) n^2), where f(·) 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(ε) n^2).
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