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 $\rho1,\dots, \rhok$. The goal is to partition the graph into $k$ disjoint sets $P1,\dots, Pk$ satisfying $|Pi|\leq \rhoi n$ so as to minimize the number of edges cut by the partition. Our algorithm has an approximation ratio of $O(\sqrt{\log n \log k})$ for general graphs, and an $O(1)$ approximation for graphs with excluded minors. This is an improvement upon the $O(\log n)$ 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 $Pi$ the vertex is assigned to. In the latter case, each vertex $u$ has a $d$-dimensional weight $r(u,i) = (r1(u,i), \dots, rd(u,i))$ if $u$ is assigned to $Pi$. Each set $Pi$ has a $d$-dimensional capacity $c(i) = (c1(i),\dots, cd(i))$. The goal is to find a partition such that $\sum{u\in {P_i}} r(u,i) \leq c(i)$ coordinate-wise.