This paper presents algorithms for computing a minimum 3-way cut and a niinimuni 4-waLy cut of an undirected weighted graph G. Let G = ( V , E ) be an undirected graph with n vertices, 7 i i edges and posit~ive edge weights. Goldschiiiidt e t al. presented an algorithm for thc niixiimum k-way rut problem with fixed k , that requires o (~~) aiid ~( n " ) maximum flow comput>ations, respect,ively, to coniput,c~ a iniiiimiim 3-way cut and a minimum 4-way cut of G. In this paper. we first show some propertics on mininiuni 3-way cuts an'd minirnum 4-way cut,s, which indicate a recursive structure of the minimuni k-way cut problem when k = 3 and 3 . Then, based on those propert,ies, we give divide-and-coiiquer algorithms for computing a niininium 3-way cut and a minimum 4-way cut of G , which require O (~L~) and O(n4) niaxiniuin flow roniputations, respectively. This means that, the proposed algorithms are the fastest ones ever known.
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