Planted Models for $k$-way Edge and Vertex Expansion

Graph partitioning problems are a central topic of study in algorithms and complexity theory. Edge expansion and vertex expansion, two popular graph partitioning objectives, seek a $2$-partition of the vertex set of the graph that minimizes the considered objective. However, for many natural applications, one might require a graph to be partitioned into $k$ parts, for some $k \geq 2$. For a $k$-partition $S_1, \ldots, S_k$ of the vertex set of a graph $G = (V,E)$, the $k$-way edge expansion (resp. vertex expansion) of $\{S_1, \ldots, S_k\}$ is defined as $\max_{i \in [k]} \Phi(S_i)$, and the balanced $k$-way edge expansion (resp. vertex expansion) of $G$ is defined as $$\min_{ \{S_1, \ldots, S_k\} \in \mathcal{P}_k} \max_{i \in [k]} \Phi(S_i) \, ,$$ where $\mathcal{P}_k$ is the set of all balanced $k$-partitions of $V$ (i.e each part of a $k$-partition in $\mathcal{P}_k$ should have cardinality $|V|/k$), and $\Phi(S)$ denotes the edge expansion (resp. vertex expansion) of $S \subset V$. We study a natural planted model for graphs where the vertex set of a graph has a $k$-partition $S_1, \ldots, S_k$ such that the graph induced on each $S_i$ has large expansion, but each $S_i$ has small edge expansion (resp. vertex expansion) in the graph. We give bi-criteria approximation algorithms for computing the balanced $k$-way edge expansion (resp. vertex expansion) of instances in this planted model.