Nonuniform graph partitioning with unrelated weights

We give a bi-criteria approximation algorithm for the Minimum Nonuniform Partitioning problem, recently introduced by Krauthgamer, Naor, Schwartz and Talwar (2014). In this problem, we are given a graph G = (V,E) on n vertices and k numbers ρ₁,..., ρ_k. The goal is to partition the graph into k disjoint sets P₁,..., P_k satisfying |P_i| ≤ ρ_in so as to minimize the number of edges cut by the partition. Our algorithm has an approximation ratio of O(√log n log k)for general graphs, and an O(1) approximation for graphs with excluded minors. This is an improvement upon the O(logn) algorithm of Krauthgamer, Naor, Schwartz and Talwar (2014). Our approximation ratio matches the best known ratio for the Minimum (Uniform) k-Partitioning problem. We extend our results to the case of "unrelated weights" and to the case of "unrelated d-dimensional weights". In the former case, different vertices may have different weights and the weight of a vertex may depend on the set P_i the vertex is assigned to. In the latter case, each vertex u has a d-dimensional weight r(u,i) = (r₁(u,i),..., r_d (u,i)) if u is assigned to P_i. Each set P_i has a d-dimensional capacity c(i) = (c₁(i),..., c_d (i)). The goal is to find a partition such that ∑_u∈Pi r(u, i) ≤ c(i) coordinate-wise.