The Complexity of Approximating Vertex Expansion

TL;DR

This paper establishes the computational difficulty of approximating vertex expansion in graphs, showing it is harder than edge expansion under the SSE hypothesis, and provides algorithms and hardness results for this problem.

Contribution

It introduces a polynomial-time algorithm for vertex expansion approximation and proves SSE-hardness, demonstrating vertex expansion's greater complexity compared to edge expansion.

Findings

Polynomial-time algorithm achieves O(√(OPT log d)) approximation.

SSE-hardness results show difficulty in approximating vertex expansion.

Vertex expansion is harder to certify than edge expansion in graphs.

Abstract

We study the complexity of approximating the vertex expansion of graphs $G = (V,E)$, defined as \[ \Phi^V := \min_{S \subset V} n \cdot \frac{|N(S)|}{|S| |V \backslash S|}. \] We give a simple polynomial-time algorithm for finding a subset with vertex expansion $O(\sqrt{OPT \log d})$ 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 $C\sqrt{OPT \log d}$ for an absolute constant $C$. In particular, this implies for all constant $\epsilon > 0$, it is SSE-hard to distinguish whether the vertex expansion $< \epsilon$ or at least an absolute constant. The analogous threshold for edge expansion is $\sqrt{OPT}$ with no dependence on the degree; 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 {\it Unique Games} instance obtained from the \SSE hypothesis to the vertex expansion problem. It involves the definition of a smoother intermediate problem we call {\sf 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.