In this paper we consider the problem of testing whether a graph has bounded
arboricity. The family of graphs with bounded arboricity includes, among
others, bounded-degree graphs, all minor-closed graph classes (e.g. planar
graphs, graphs with bounded treewidth) and randomly generated preferential
attachment graphs. Graphs with bounded arboricity have been studied extensively
in the past, in particular since for many problems they allow for much more
efficient algorithms and/or better approximation ratios.
We present a tolerant tester in the sparse-graphs model. The sparse-graphs
model allows access to degree queries and neighbor queries, and the distance is
defined with respect to the actual number of edges. More specifically, our
algorithm distinguishes between graphs that are ϵ-close to having
arboricity α and graphs that c⋅ϵ-far from having
arboricity 3α, where c is an absolute small constant. The query
complexity and running time of the algorithm are
O~(mn⋅ϵlog(1/ϵ)+mn⋅α⋅(ϵ1)O(log(1/ϵ))) where n denotes
the number of vertices and m denotes the number of edges. In terms of the
dependence on n and m this bound is optimal up to poly-logarithmic factors
since Ω(n/m) queries are necessary (and α=O(m)).
We leave it as an open question whether the dependence on 1/ϵ can be
improved from quasi-polynomial to polynomial. Our techniques include an
efficient local simulation for approximating the outcome of a global (almost)
forest-decomposition algorithm as well as a tailored procedure of edge
sampling