Testing Conductance in General Graphs

In this paper, we study the problem of testing the conductance of a given graph in the general graph model. Given distance parameter ε and any constant σ > 0, there exists a tester whose running time isO( (1+σ)/2·logn·log ε ε·Φ2 ), where n is the number of vertices andm is the number of edges of the input graph. With probability at least 2/3, the tester accepts all graphs of conductance at least Φ, and rejects any graph that is ε-far from any graph of conductance at least α′ for α′ = Ω(Φ). This result matches the best testing algorithm for the bounded degree graph model in [5]. Our main technical contribution is the non-uniform Zig-Zag product, which generalizes the standard Zig-Zag product given by Reingold et. al. [9] to the unregular case. It converts any graph to a regular one and keeps (roughly) the size and conductance, by choosing a proper Zig-Zag graph sequence. This makes it easy to test the conductance of the given graph on the new one. The analysis and applications of non-uniform Zig-Zag product may be independently interesting.