  The separability of clusters is one of the most desired properties in
clustering. There is a wide range of settings in which different clusterings of
the same data set appear. We are interested in applications where there is a
need for an explicit, gradual transition of one separable clustering into
another one. This transition should be a sequence of simple, natural steps that
upholds separability of the clusters throughout.
  We design an algorithm for such a transition. We exploit the intimate
connection of separability and linear programming over bounded-shape partition
and transportation polytopes: separable clusterings lie on the boundary of
partition polytopes, form a subset of the vertices of the corresponding
transportation polytopes, and circuits of both polytopes are readily
interpreted as sequential or cyclical exchanges of items between clusters. This
allows for a natural approach to achieve the desired transition through a
combination of two walks: an edge walk between two so-called radial clusterings
in a transportation polytope, computed through an adaptation of classical tools
of sensitivity analysis and parametric programming; and a walk from a separable
clustering to a corresponding radial clustering, computed through a tailored,
iterative routine updating cluster sizes and re-optimizing the cluster
assignment of items.
