  Clustering is one of the fundamental tasks in data analytics and machine
learning. In many situations, different clusterings of the same data set become
relevant. For example, different algorithms for the same clustering task may
return dramatically different solutions. We are interested in applications in
which one clustering has to be transformed into another; e.g., when a gradual
transition from an old solution to a new one is required. In this paper, we
devise methods for constructing such a transition based on linear programming
and network theory. We use a so-called clustering-difference graph to model the
desired transformation and provide methods for decomposing the graph into a
sequence of elementary moves that accomplishes the transformation. These moves
are equivalent to the edge directions, or circuits, of the underlying partition
polytopes. Therefore, in addition to a conceptually new metric for measuring
the distance between clusterings, we provide new bounds on the circuit diameter
of these partition polytopes.
