Shared-memory Exact Minimum Cuts

TL;DR

This paper presents a highly optimized, parallel exact algorithm for the minimum cut problem in graphs, significantly outperforming previous state-of-the-art methods through advanced reductions, data structures, and bounds.

Contribution

It introduces a new, faster exact minimum cut algorithm that integrates inexact bounds, improved reductions, and parallel processing to outperform existing solvers.

Findings

The new algorithm is significantly faster than previous methods.

Parallelization improves contraction routine efficiency.

Enhanced data structures contribute to overall speedup.

Abstract

The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. In this paper, we engineer the fastest known exact algorithm for the problem. State-of-the-art algorithms like the algorithm of Padberg and Rinaldi or the algorithm of Nagamochi, Ono and Ibaraki identify edges that can be contracted to reduce the graph size such that at least one minimum cut is maintained in the contracted graph. Our algorithm achieves improvements in running time over these algorithms by a multitude of techniques. First, we use a recently developed fast and parallel \emph{inexact} minimum cut algorithm to obtain a better bound for the problem. Then we use reductions that depend on this bound, to reduce the size of the graph much faster than previously possible. We use improved data structures to further improve the running time of our algorithm. Additionally, we parallelize the contraction routines of Nagamochi, Ono and Ibaraki. Overall, we arrive at a system that outperforms the fastest state-of-the-art solvers for the \emph{exact} minimum cut problem significantly.