Canonical Cut Hierarchy in Almost-Linear Time via Reduction to Max-Flow

Abstract

The canonical cut hierarchy recursively partitions a graph along Tutte-Nash-Williams partitions, choosing the partition with the most number of parts at each step. We give a randomized almost linear-time algorithm for computing this hierarchy in undirected graphs with polynomially bounded capacities via reduction to max-flow. This yields randomized almost linear-time algorithms for computing partition connectivity, strength, both arboricity and fractional arboricity, both integral and fractional spanning tree packing numbers, and implicit representations of the principal partition sequence and a maximum fractional spanning tree packing.