Min sum clustering with penalties

Abstract

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.

Modelling antimicrobial prescriptions in Scotland: A spatio-temporal clustering approach

“…Graph clustering looks for clusters of vertices in graphs in such a way that there are many edges within a cluster and relatively few between them (Benati et al, 2017;Schaeffer, 2007). One group of algorithms in this area are spectral methods, where an eigenvector or a combination of eigenvectors is used for computing the similarity of clusters (Nascimento & De Carvalho, 2011;Schaeffer, 2007), while another approach is to formulate a min-sum problem which minimises the sum of distances between points in a cluster (Hassin & Or, 2010). Spatial clustering describes using cluster analysis to cluster data points in geographic regions.…”