An Improved Divide-and-Conquer Algorithm for Finding All Minimum k-Way Cuts

Abstract

Given a positive integer k and an edge-weighted undirected graph G = (V,E;w), the minimum k -way cut problem is to find a subset of edges of minimum total weight whose removal separates the graph into k connected components. This problem is a natural generalization of the classical minimum cut problem and has been well-studied in the literature. A simple and natural method to solve the minimum k-way cut problem is the divide-and-conquer method: getting a minimum k-way cut by properly separating the graph into two small graphs and then finding minimum k'-way cut and k''-way cut respectively in the two small graphs, where k' + k'' = k. In this paper, we present the first algorithm for the tight case of $k'=\lfloor k/2\rfloor$. Our algorithm runs in $O(n^{4k-\lg k})$ time and can enumerate all minimum k-way cuts, which improves all the previously known divide-and-conquer algorithms for this problem.