The Small Set Vertex Expansion Problem

Abstract

Given a graph $G=(V,E)$, the vertex expansion of a set $S\subset V$ is defined as $\varPhi ^V(S)=\frac{|N(S)|}{|S|}.$ In the Small Set Vertex Expansion (SSVE) problem, we are given a graph $G=(V,E)$ and a positive integer $k\le \frac{|V(G)|}{2}$, the goal is to return a set $S\subset V(G)$ of k nodes minimizing the vertex expansion $\varPhi ^V(S)=\frac{|N(S)|}{k}$; equivalently minimizing |N(S)|. SSVE has not been as well studied as its edge-based counterpart Small Set Expansion (SSE). SSE, and SSVE to a less extend, have been studied due to their connection to other hard problems including the Unique Games Conjecture and Graph Colouring. Using the hardness of Minimum k-Union problem, we prove that Small Set Vertex Expansion problem is NP-complete. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[1]-hard when parameterized by k, (2) the problem is fixed-parameter tractable (FPT) when parameterized by the neighbourhood diversity nd, and (3) it is fixed-parameter tractable (FPT) when parameterized by treewidth tw of the input graph.

The author’s research was supported in part by the Science and Engineering Research Board (SERB), Govt. of India, under Sanction Order No. MTR/2018/001025.