Ultra-fast approximation of minimum cut by condensing traversal trees in large-scale graphs

Minimum cut (min-cut) algorithms are of great significance in numerous applications, making the need for rapid solutions highly pressing. Current acceleration techniques mainly focus on two aspects: optimizing the algorithm’s logical structure and reducing the size of graph data. Considering that each traversal tree corresponds to a particular cut, we approach the problem from the perspective of cut enumeration. Our proposed algorithm conducts cut enumeration by leveraging the depth-first traversal tree. For each node, it identifies the optimal tree with the smallest local cut and then performs condensing operations to achieve acceleration. After condensing, any pair of nodes can be separated by the condensed trees that contain only one of them. In a graph having $M$ edges, the time complexity of the preprocessing step in the serial algorithm version is as low as $O\left(M\right)$. It can accurately determine the min-cut for over 99.9% of node pairs. The calculation time for each pair is remarkably short, being only a thousandth of that of the most efficient existing methods. Furthermore, when dealing with a graph having tens of millions of edges on a common computing node, the time consumption is just a few microseconds. As a result, this algorithm can serve as an effective heuristic method for min-cut approximation.