A Distributed Mincut/Maxflow Algorithm Combining Path Augmentation and Push-Relabel

Abstract

We present a novel distributed algorithm for the minimum s-t cut problem, suitable for solving large sparse instances. Assuming vertices of the graph are partitioned into several regions, the algorithm performs path augmentations inside the regions and updates of the push-relabel style between the regions. The interaction between regions is considered expensive (regions are loaded into the memory one-by-one or located on separate machines in a network). The algorithm works in sweeps, which are passes over all regions. Let $\mathcal B$ be the set of vertices incident to inter-region edges of the graph. We present a sequential and parallel versions of the algorithm which terminate in at most $2|{\mathcal B}|^2+1$ sweeps. The competing algorithm by Delong and Boykov uses push-relabel updates inside regions. In the case of a fixed partition we prove that this algorithm has a tight O(n²) bound on the number of sweeps, where n is the number of vertices. We tested sequential versions of the algorithms on instances of maxflow problems in computer vision. Experimentally, the number of sweeps required by the new algorithm is much lower than for the Delong and Boykov’s variant. Large problems (up to 10⁸ vertices and 6·10⁸ edges) are solved using under 1GB of memory in about 10 sweeps.