Testing Expansion in Bounded-Degree Graphs

Abstract

We consider the problem oftesting expansion in bounded-degree graphs. We focus on the notion ofvertex expansion: an α-expander is a graphG= (V,E) in which every subsetU⊆Vof at most |V|/2 vertices has a neighbourhood of size at least α ⋅ |U|. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time$\widetilde{\O}(\sqrt{n})$. We prove that the property-testing algorithm proposed by Goldreich and Ron with appropriately set parameters accepts every α-expander with probability at least$\frac23$and rejects every graph that is ϵ-far from any α*-expander with probability at least$\frac23$, where$\expand^* \,{=}\, \Theta(\frac{\expand^2}{d^2 \log(n/\epsilon)})$anddis the maximum degree of the graphs. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is$\O(\frac{d^2 \sqrt{n} \log(n/\epsilon)} {\expand^2 \epsilon^3})$.