Min sum clustering with penalties

Given a complete graph G = (V, E), a weight function w : E → N₀ on its edges, and p → N₀ a penalty function on its vertices, the penalized k-min-sum problem is the problem of finding a partition of V to k + 1 sets, S₁, ..., S_{k + 1}, minimizing ∑_{i = 1}^k w (S_i) + p (S_{k + 1}), where for S ⊆ V w (S) = ∑_{e = {i, j} ⊆ S} w_e, and p (S) = ∑_{i ∈ S} p_i. Our main result is a randomized approximation scheme for the metric version of the penalized 1-min-sum problem, when the ratio between the minimal and maximal penalty is bounded. For the metric penalized k-min-sum problem where k is a constant, we offer a 2-approximation.