A new–old algorithm for minimum-cut and maximum-flow in closure graphs

We present an algorithm for solving the minimum-cut problem on closure graphs without maintaining flow values. The algorithm is based on an optimization algorithm for the open-pit mining problem that was presented in 1964 (and published in 1965) by Lerchs and Grossmann. The Lerchs–Grossmann algorithm (LG algorithm) solves the maximum closure which is equivalent to the minimum-cut problem. Yet, it appears substantially different from other algorithms known for solving the minimum-cut problem and does not employ any concept of flow. Instead, it works with sets of nodes that have a natural interpretation in the context of maximum closure in that they have positive total weight and are closed with respect to some subgraph. We describe the LG algorithm and study its features and the new insights it reveals for the maximum-closure problem and the maximum-flow problem. Specifically, we devise a linear time procedure that evaluates a feasible flow corresponding to any iteration of the algorithm. We show that while the LG algorithm is pseudopolynomial, our variant algorithms have complexity of O(mn log n), where n is the number of nodes and m is the number of arcs in the graph. Modifications of the algorithm allow for efficient sensitivity and parametric analysis also running in time O(mn log n).