Approximating the Arboricity in Sublinear Time

TL;DR

This paper presents a sublinear time algorithm for approximating the arboricity of a graph with high probability, achieving near-optimal query complexity relative to the graph's arboricity.

Contribution

The authors introduce a novel sublinear time algorithm that approximates graph arboricity within a logarithmic factor, with optimal query complexity up to polylogarithmic factors.

Findings

Algorithm achieves approximation within a factor of $c ext{log}^2 n$ with high probability.

Expected query complexity is $O(n/ ext{arb}(G)) ext{poly}( ext{log} n)$, which is near-optimal.

The method extends the understanding of sublinear algorithms for graph properties.

Abstract

We consider the problem of approximating the arboricity of a graph $G= (V,E)$, which we denote by $\mathsf{arb}(G)$, 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 $\hat{\alpha}$, such that with probability $1-1/\textrm{poly}(n)$, $\mathsf{arb}(G)/c\log^2 n \leq \hat{\alpha} \leq \mathsf{arb}(G)$, where $n=|V|$ and $c$ is a constant. The expected query complexity and running time of the algorithm are $O(n/\mathsf{arb}(G))\cdot \textrm{poly}(\log n)$, and this upper bound also holds with high probability. %($\widetilde{O}(\cdot)$ is used to suppress $\textrm{poly}(\log n)$ dependencies). This bound is optimal for such an approximation up to a $\textrm{poly}(\log n)$ factor.