Approximation algorithms for minimum K-cut

Let G = (V, E) be a complete undirected graph, with node set V = {v₁,...,v_n} and edge set E. The edges (v_i, v_j) ε E have nonnegative weights that satisfy the triangle inequality. Given a set of integers K = {k_i}_i=t^p (∑_i=1^pk_i≤ |V|), the minimum K-cutproblem is to compute disjoint subsets with sizes {k_i}_i=1^p, minimizing the total weight of edges whose two ends are in different subsets. We demonstrate that for any fixed p it is possible to obtain in polynomial time an approximation of at most three times the optimal value. We also prove bounds on the ratio between the weights of maximum and minimum cuts.