Testing Bounded Arboricity

In this article, we consider the problem of testing whether a graph has bounded arboricity. The class of graphs with bounded arboricity includes many important graph families (e.g., planar graphs and randomly generated preferential attachment graphs). Graphs with bounded arboricity have been studied extensively in the past, particularly because for many problems, they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the general-graphs model. The general-graphs model allows access to degree and neighbor queries, and the distance is defined with respect to the actual number of edges. Namely, we say that a graph G is ϵ-close to having arboricity α if by removing at most an ϵ-fraction of its edges, we can obtain a graph G′ that has arboricity α, and otherwise we say that G is ϵ-far. Our algorithm distinguishes between graphs that are ϵ-close to having arboricity α and graphs that are c á ϵ-far from having arboricity 3α, where c is an absolute small constant. The query complexity and running time of the algorithm are Õ (n / ϵm) + (1 / ϵ)^O(log(1/ϵ)), where n denotes the number of vertices and m denotes the number of edges (we use the notation Õ to hide poly-logarithmic factors in n). 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.