The Complexity of Approximating Vertex Expansion

We study the complexity of approximating the vertex expansion of graphs G = (V, E), defined as Φ^V def = minSCV n . |N(S)|/(|S||VS). We give a simple polynomialtime algorithm for finding a subset with vertex expansion O(√(Φ^V log d)) where d is the maximum degree of the graph. Our main result is an asymptotically matching lower bound: under the Small Set Expansion (SSE) hypothesis, it is hard to find a subset with expansion less than C(√(Φ^V log d)) for an absolute constant C. In particular, this implies for all constant ε > 0, it is SSE-hard to distinguish whether the vertex expansion <; ε or at least an absolute constant. The analogous threshold for edge expansion is √Φ with no dependence on the degree (Here Φ denotes the optimal edge expansion). Thus our results suggest that vertex expansion is harder to approximate than edge expansion. In particular, while Cheeger´s algorithm can certify constant edge expansion, it is SSE-hard to certify constant vertex expansion in graphs.