A faster parametric minimum-cut algorithm

Abstract

Galloet al. [4] recently examined the problem of computing on line a sequence ofk maximum flows and minimum cuts in a network ofn nodes, where certain edge capacities change between each flow. They showed that for an important class of networks, thek maximum flows and minimum cuts can be computed simply in O(n³+kn²) total time, provided that the capacity changes are made “in order.” Using dynamic trees their time bound isO(nm log(n²/m)+km log(n²/m)). We show how to reduce the total time, using a simple algorithm, to O(n³+kn) for explicitly computing thek minimum cuts and implicitly representing thek flows. Using dynamic trees our bound is O(nm log(n²/m)+kn log(n²/m)). We further reduce the total time toO(n ²√m) ifk is at most O(n). We also apply the ideas from [10] to show that the faster bounds hold even when the capacity changes are not “in order,” provided we only need the minimum cuts; if the flows are required then the times are respectively O(n³+km) and O(n²√m). We illustrate the utility of these results by applying them to therectilinear layout problem.