Non-Uniform Graph Partitioning [chapter]

Robert Krauthgamer, Joseph (Seffi) Naor, Roy Schwartz, Kunal Talwar
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms  
We consider the problem of NON-UNIFORM GRAPH PAR-TITIONING, where the input is an edge-weighted undirected graph G = (V, E) and k capacities n 1 , . . . , n k , and the goal is to find a partition {S 1 , S 2 , . . . , S k } of V satisfying |S j | ≤ n j 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
more » ... ION, MINIMUM BALANCED CUT, and MINIMUM k-PARTITIONING. Unlike these problems, NON-UNIFORM 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.
doi:10.1137/1.9781611973402.91 dblp:conf/soda/KrauthgamerNST14 fatcat:koi5mdoee5aohckex7hxuu5whq