Faster algorithms for finding a minimum K-way cut in a weighted graph

This paper presents algorithms for computing a minimum 3-way cut and a minimum 4-way cut of an undirected weighted graph G. Let G=(V,E) be an undirected graph with n vertices, m edges and positive edge weights. Goldschmidt et al. presented an algorithm for the minimum κ-way cut problem with fixed κ, that requires O(n⁴) and O(n⁹) maximum flow computations, respectively, to compute a minimum 3-way cut and a minimum 4-way cut of G. In this paper, we first show some properties on minimum 3-way cuts and minimum 4-way cuts, which indicate a recursive structure of the minimum X-way cut problem when κ=3 and 4. Then, based on those properties, we give divide-and-conquer algorithms for computing a minimum 3-way cut and a minimum 4-way cut of G, which require O(n³ ) and O(n⁴) maximum flow computations, respectively. This means that the proposed algorithms are the fastest ones ever known