Min sum clustering with penalties

Traditionally, clustering problems are investigated under the assumption that all objects must be clustered. A shortcoming of this formulation is that a few distant objects, called outliers, may exert a disproportionately strong influence over the solution. In this work we investigate the k-MIN-SUM clustering problem while addressing outliers in a meaningful way. Given a complete graph G = (V, E), a weight function w: E → IN₀ on its edges, and p → IN₀ a penalty function on its nodes, the PENALIZED k-MIN-SUM PROBLEM is the problem of finding a partition of V to k + 1 sets, {S₁,..., S_k+1), minimizing Σ _i=1^k w(S_i)+p(S_k+1), where for S ⊆ V w(S) = Σ_{e={i, j}⊂}S w_e, and p(S) = Σ_iεS p_i. We offer an efficient 2-approximation to the penalized 1-min-sum problem using a primal-dual algorithm. We prove that the penalized 1-min-sum problem is NP-hard even if tu is a metric and present a randomized approximation scheme for it. For the metric penalized k-min-sum problem we offer a 2-approximation.