Two recent algorithms for the global minimum cut problem

• Authors: Ashok Subramanian
• Venue: ACM SIGACT News
• Year: 1995

This article provides a self-contained exposition of two recent algorithms for the Global Minimum Cut problem. In this problem, we are given an undirected graph with positive weights on its edges, and we seek a proper subset S of the vertices that minimizes the total weight of all edges that have exactly one endpoint in S. Unlike traditional approaches to this problem, the new algorithms are not based on network flow computations. Instead they are based on a decomposition of the weighted graph into a sequence of weighted forests. The decomposition is used to identify a light set of edges that contains every light cutset of the graph. In general, such a decomposition requires too much space (and hence time) to write down. However, there is an efficient graph search procedure, Maximum Adjacency Search, that can be used to compute an implicit representation of such a decomposition in O(m + n log n) time.The first algorithm we discuss, due to Nagamochi and Ibaraki [5, 6], requires O(n) invocations of Maximum Adjacency Search to compute a global minimum cut. The second algorithm, due to Matula [4], requires O(log n) invocations of Maximum Adjacency Search to compute an approximately minimum cut. We do not discuss here the randomized algorithms of Karger or the matroid-based approach of Gabow.