Approximating the Arboricity in Sublinear Time

10/28/2021

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by   Talya Eden, et al.

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We consider the problem of approximating the arboricity of a graph G= (V,E), which we denote by 𝖺𝗋𝖻(G), in sublinear time, where the arboricity of a graph is the minimal number of forests required to cover its edges. An algorithm for this problem may perform degree and neighbor queries, and is allowed a small error probability. We design an algorithm that outputs an estimate α̂, such that with probability 1-1/poly(n), 𝖺𝗋𝖻(G)/clog^2 n ≤α̂≤𝖺𝗋𝖻(G), where n=|V| and c is a constant. The expected query complexity and running time of the algorithm are O(n/𝖺𝗋𝖻(G))·poly(log n), and this upper bound also holds with high probability. poly(log n) dependencies). This bound is optimal for such an approximation up to a poly(log n) factor.