In this paper we study the capacitated vertex cover problem, a generalization of the well-known vertex cover problem. Given a graph G = (V , E) with weights on the vertices, the goal is to cover all the edges by picking a cover of minimum weight from the vertices. When we pick a copy of a vertex, we pay the weight of the vertex and cover up to a pre-specified number of edges incident on this vertex (its capacity). The problem is NP-hard. We give a primal-dual based approximation algorithm with an approximation guarantee of 2, and study several generalizations, as well as the problem restricted to trees.

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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 → IN 0 on its edges, and p → IN 0 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, S 1 , . . . , S k+1 , minimizing k i=1 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.

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 → IN 0 on its edges, and p → IN 0 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, S 1 , . . . , S k+1 , minimizing, 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.