Non-Uniform Graph Partitioning

We consider the problem of NON-UNIFORM GRAPH PARTITIONING, where the input is an edge-weighted undirected graph G = (V,E) and k capacities n1, . . . , nk, and the goal is to find a partition {S1, S2, . . . , Sk} of V satisfying |Sj | ≤ nj for all 1 ≤ j ≤ k, that minimizes the total weight of edges crossing between different parts. This natural graph partitioning problem arises in practical scenarios, and generalizes well-studied balanced partitioning problems such as MINIMUM BISECTION, MINIMUM BALANCED CUT, and MINIMUM k-PARTITIONING. Unlike these problems, NONUNIFORM GRAPH PARTITIONING seems to be resistant to many of the known partitioning techniques, such as spreading metrics, recursive partitioning, and Räcke’s tree decomposition, because k can be a function of n and the capacities could be of different magnitudes. We present a bicriteria approximation algorithm for NON-UNIFORM GRAPH PARTITIONING that approximates the objective within O(log n) factor while deviating from the required capacities by at most a constant factor. Our approach is to apply stopping-time based concentration results to a simple randomized rounding of a configuration LP. These concentration bounds are needed as the commonly used techniques of bounded differences and bounded conditioned variances do not suffice.