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

TL;DR

This paper introduces a planted model for graphs with a $k$-partition where each part has high internal expansion but low expansion between parts, and provides algorithms to approximate the $k$-way expansion in such graphs.

Contribution

The paper proposes a natural planted model for $k$-partitioned graphs and develops bi-criteria approximation algorithms for the $k$-way expansion problem within this model.

Findings

Algorithms achieve approximation guarantees in the planted model.

The model captures realistic graph partitioning scenarios.

Provides theoretical analysis of algorithm performance.

Abstract

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.