Testing expansion in bounded-degree graphs

  • Authors:

  • Artur Czumaj;Christian Sohler

  • Affiliations:

  • Department of computer science and centre for discrete mathematics and its applications (dimap), university of warwick, coventry cv4 7al, uk (e-mail: a.czumaj@warwick.ac.uk);Department of computer science, technische universität dortmund, d-44221 dortmund, germany (e-mail: christian.sohler@tu-dortmund.de)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2010

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Abstract

We consider the problem of testing expansion in bounded-degree graphs. We focus on the notion of vertex expansion: an α-expander is a graph G = (V, E) in which every subset U ⊆ V of 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 $\expand^* \,{=}\, \Theta(\frac{\expand^2}{d^2 \log(n/\epsilon)})$ and d is the maximum degree of the graphs. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is $\O(\frac{d^2 \sqrt{n} \log(n/\epsilon)} {\expand^2 \epsilon^3})$.