Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm

Computing a minimum s-t cut in a graph is a solution to a wide range of computer vision problems, and is often done using the Boykov-Kolmogorov (BK) algorithm. In this paper, we revisit the BK algorithm from both a theoretical and practical point of view. We improve the analysis of the time complexity of the BK algorithm to O(mn|C|) and propose a new algorithm, the fast and compact BK (fcBK) algorithm, with a time complexity of O(m|C|), where m, n, and |C| are the number of edges, number of vertices, and the capacity of the cut, respectively. We additionally propose a compact graph representation that allows our implementation to find a minimum s-t cut in a graph with upwards of 109 vertices and 1010 edges on a machine with 128 GB of memory. We find our implementation of the BK algorithm to be the fastest available implementation of the BK algorithm when evaluating on a comprehensive set of benchmark datasets, highlighting the importance of memory-efficient implementations. We make our implementations publicly available for further research and implementation development within minimum s-t cut algorithms.