  In this paper, we study several generalizations of multiway cut where the
terminals can be chosen as \emph{representatives} from sets of
\emph{candidates} $T_1,\ldots,T_q$. In this setting, one is allowed to choose
these representatives so that the minimum-weight cut separating these sets
\emph{via their representatives} is as small as possible. We distinguish
different cases depending on (A) whether the representative of a candidate set
has to be separated from the other candidate sets completely or only from the
representatives, and (B) whether there is a single representative for each
candidate set or the choice of representative is independent for each pair of
candidate sets. For fixed $q$, we give approximation algorithms for each of
these problems that match the best known approximation guarantee for multiway
cut. Our technical contribution is a new extension of the CKR relaxation that
preserves approximation guarantees. For general $q$, we show $o(\log
q)$-inapproximability for all cases where the choice of representatives may
depend on the pair of candidate sets, as well as for the case where the goal is
to separate a fixed node from a single representative from each candidate set.
As a positive result, we give a $2$-approximation algorithm for the case where
we need to choose a single representative from each candidate set. This is a
generalization of the $(2-2/k)$-approximation for k-cut, and we can solve it by
relating the tree case to optimization over a gammoid.
