  We propose a new variant of the k-median problem, where the objective
function models not only the cost of assigning data points to cluster
representatives, but also a penalty term for disagreement among the
representatives. We motivate this novel problem by applications where we are
interested in clustering data while avoiding selecting representatives that are
too far from each other. For example, we may want to summarize a set of news
sources, but avoid selecting ideologically-extreme articles in order to reduce
polarization.
  To solve the proposed k-median formulation we adopt the local-search
algorithm of Arya et al. We show that the algorithm provides a provable
approximation guarantee, which becomes constant under an assumption on the
minimum number of points for each cluster. We experimentally evaluate our
problem formulation and proposed algorithm on datasets inspired by the
motivating applications. In particular, we experiment with data extracted from
Twitter, the US Congress voting records, and popular news sources. The results
show that our objective can lead to choosing less polarized groups of
representatives without significant loss in representation fidelity.
