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 ≥ 2. For a k-partition S1, ldots, Sk of the vertex set of a graph G = (V,E), the k-way edge expansion (resp. vertex expansion) of S1, ldots, Sk is defined as maxi ∈ [k] Phi(Si), and the balanced k-way edge expansion (resp. vertex expansion) of G is defined as [ min_ S1, ldots, Sk ∈ Pk maxi ∈ [k] Phi(Si) , , ] where Pk is the set of all balanced k-partitions of V (i.e each part of a k-partition in Pk 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 S1, ldots, Sk such that the graph induced on each Si has large expansion, but each Si 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.