# 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 S_1, ..., S_k of the vertex set of a graph G = (V,E), the k-way edge expansion (resp. vertex expansion) of {S_1, ..., S_k} is defined as max_i ∈ [k]Φ(S_i), and the balanced k-way edge expansion (resp. vertex expansion) of G is defined as min_{S_1, ..., S_k}∈P_kmax_i ∈ [k]Φ(S_i) , where P_k is the set of all balanced k-partitions of V (i.e each part of a k-partition in P_k should have cardinality |V|/k), and Φ(S) denotes the edge expansion (resp. vertex expansion) of S ⊂ V. We study a natural planted model for graphs where the vertex set of a graph has a k-partition S_1, ..., 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.

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