The Complexity of Approximating Vertex Expansion

Abstract

We study the complexity of approximating the vertex expansion of graphs G = (V, E), defined asWe give a simple polynomial-time 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.Our proof is via a reduction from the Unique Games instance obtained from the SSE hypothesis to the vertex expansion problem. It involves the definition of a smoother intermediate problem we call Balanced Analytic Vertex Expansion which is representative of both the vertex expansion and the conductance of the graph. Both reductions (from the UGC instance to this problem and from this problem to vertex expansion) use novel proof ideas.

Integrality Gaps and Approximation Algorithms for Dispersers and Bipartite Expanders

“…To the best of our knowledge, all known reductions [RST12,LRV13] from the Small Set Expansion hypothesis only work for dictatorship tests with small noise, but the dictatorship test mentioned above with completeness 1 − 1−ρ ρd+1−ρ and soundness 1 − (1 − ρ) d requires pairwise independence. Because such a reduction from SSE to the disperser problem would provide an explicit construction matching the construction from the probabilistic method, it is interesting to discover more reductions from the Small Set Expansion hypothesis that support more dictatorship tests, which include tests with pairwise independence.…”

“…At the same time, they also provided an efficient algorithm based on semidefinite programmings with an asymptotic matching approximation ratio that given a graph with vertex expansion ǫ and bounded degree d, finds a subset with vertex expansion O( √ ǫ log d). When the vertex expansion in expanders is independent of the left degree, we prove that it is SSE-hard to distinguish between good expanders and bad expanders when ρ is small enough and degree is large enough by amplifying the gap in the hardness result of [LRV13](see Theorem 5.3 for a formal statement). In another extreme case that the bipartite graph G has a ρN -subset with at most (1+ ǫ)ρM neighbor, We provide an efficient algorithm with an asymptotic matching approximation ratio by following the previous work of [LRV13, CMM06, BFK + 11, LM14].…”

“…on the approximation ratio compared to the integrality gap, whose loss log d is from the preprocessor and min{ For (ρN, ρ(1 + ǫ)M )-dispersers, the hardness result is based on the work of Louis, Raghavendra and Vempala [LRV13], which provide a basic hardness result like 1 + ǫ and 1 + √ ǫ log d. Then we amplify the gap using graph products by enlarging the degrees in the bipartite graph. For the approximation algorithm, we follow the approach of Louis and Makarychev [LM14] and apply the idea of finding balanced cut by the sparsest cut algorithm because the expansion of the graph is very small in this case.…”

“…It is known that the approximation of Max Bisection cannot be better than Max Cut (the reduction is to make two copies of the graph), however, the best approximation ratio of Max Bisection is 0.8776 [ABG13] to our best knowledge, which is slightly smaller than the approximation ratio of Max Cut 0.8786 [GW95]. In a graph H = (V, E) that is not necessarily bipartite, it is more interesting to consider the vertices in V \ S connected to S, which is Γ(S) \ S, and define the vertex expansion of S to be |V | · |Γ(S)\S| |S|·|V \S| in H. Recently, Louis, Raghavendra and Vempala [LRV13] showed that vertex expansion is much harder to approximate than edge expansion, which is easy to approximate by Cheeger's inequality from the second eigenvalue. They proved that it is SSE-hard to determine whether the vertex expansion of a given graph is at most O(ǫ) or at least Ω( √ ǫ log d) for small enough ǫ.…”

“…Because at most O( N k 1.5 ) vertices are added to T , with constant probability, We present our results for (ρN, ρ(1 + ǫ)M )-dispersers when ǫ is small enough. We first make a reduction from vertex expansion [LRV13] to the disperser problem which gives a hardness result based on Small-Set Expansion hypothesis. Then we show there is a polynomial time algorithm that has an approximation ratio close to the hardness result when d|D.…”