Testing Expansion in Bounded Degree Graphs

We consider the problem of testing graph expansion in the bounded degree model. We give a property tester that given a graph with degree bound d, an expansion bound α, and a parameter ε > 0, accepts the graph with high probability if its expansion is more than α, and rejects it with high probability if it is ε-far from any graph (with degree bound 2d) with expansion Ω(α). The algorithm runs in time Õ( 0.5+μ d 2 εα2 ) for any given constant μ > 0. 1 Property Testing of Expansion We are given an input graph G = (V,E) on n vertices with degree bound d. Assume that d is a sufficiently large constant. Given a cut (S, S̄) (where S̄ = V \ S) in the graph, let E(S, S̄) be the number of edges crossing the cut. The expansion of the cut is E(S,S̄) min{|S|,|S̄|} . The expansion of the graph, αG, is the expansion of the minimum expansion cut in the graph. We can estimate graph expansion using random walks. Consider the following slight modification of the standard random walk on the graph: starting from any vertex, the probability of choosing any outgoing edge is 1/2d, and with the remaining probability, the random walk stays at the current node. Thus, for a vertex of degree d′ ≤ d, the probability of a self-loop is 1−d′/2d ≥ 1/2. This walk is symmetric and reversible; therefore, its stationary distribution is uniform over the entire graph. Thus, the conductance of a cut (S, S̄) in the graph is exactly its expansion divided by 2d. The conductance of the graph, ΦG, is the conductance of the minimum conductance cut in the graph. Thus, ΦG = αG/2d. The graph is represented by an adjacency list, so we have constant time access to the neighbors of any vertex. The tester is given two parameters Φ and ε. Each of these will be assumed to be at most some sufficiently small constant. The tester must (with high probability) accept if ΦG > Φ and reject if G is ε-far from having ΦG > cΦ 2 (for some absolute constant c), even when the degree is allowed to be 2d. This means that G has to be changed at εdn edges (either removing or adding) to make the conductance > cΦ2 keeping the degrees to be at most 2d. Note that this formulation in terms of the conductance is equivalent to the property testing problem given in the abstract, since the conductance and expansion in the graph are related by scaling. We can always assume that ε ≤ 1/4d. If not, then we can simply set ε to be 1/4d in our tester and this will not affect the results. This formulation was first considered by Goldreich and Ron [2], who described an approach towards designing the required property tester. They proposed an algorithm, but analysis relied on an unproven combinatorial conjecture. Note that we solve a weaker version of their problem, because we only reject graphs that are far from being expanders with degree bound 2d (instead of d). Our algorithm uses the same ideas as their paper, but we use different algebraic techniques to 1 Electronic Colloquium on Computational Complexity, Report No. 76 (2007)