  Let $\mathcal{P}$ and $\mathcal{P}'$ be finite partitions of the set $V$.
Finding good correspondences between the parts of $\mathcal{P}$ and those of
$\mathcal{P}'$ is helpful in classification, pattern recognition, and network
analysis. Unlike common similarity measures for partitions that yield only a
single value, we provide specifics on how $\mathcal{P}$ and $\mathcal{P'}$
correspond to each other.
  To this end, we first define natural collections of best correspondences
under three constraints \cone, \ctwo, and \cthree. In case of \cone, the best
correspondences form a minimum cut basis of a certain bipartite graph, whereas
the other two lead to minimum cut bases of $\mathcal{P}$ \wrt $\mathcal{P}'$.
We also introduce a constraint, \cfour, which tightens \cthree; both are useful
for finding consensus partitions. We then develop branch-and-bound algorithms
for finding minimum $P_s$-$P_t$ cuts of $\mathcal{P}$ and thus $\vert
\mathcal{P} \vert -1$ best correspondences under \ctwo, \cthree, and \cfour,
respectively.
  In a case study, we use the correspondences to gain insight into a community
detection algorithm. The results suggest, among others, that only very minor
losses in the quality of the correspondences occur if the branch-and-bound
algorithm is restricted to its greedy core. Thus, even for graphs with more
than half a million nodes and hundreds of communities, we can find hundreds of
best or almost best correspondences in less than a minute.
