  In this work, we first study a natural generalisation of the Min-Cut problem,
where a graph is augmented by a superadditive set function defined on its
vertex subsets. The goal is to select a vertex subset such that the weight of
the induced cut plus the set function value are minimised. In addition, a lower
and upper bound is imposed on the solution size. We present a polynomial-time
algorithm for enumerating all near-optimal solutions of this Bounded
Generalised Min-Cut problem.
  Second, we apply this novel algorithm to surjective general-valued constraint
satisfaction problems (VCSPs), i.e., VCSPs in which each label has to be used
at least once. On the Boolean domain, Fulla, Uppman, and Zivny [ACM ToCT'18]
have recently established a complete classification of surjective VCSPs based
on an unbounded version of the Generalised Min-Cut problem. Their result
features the discovery of a new non-trivial tractable case called EDS that does
not appear in the non-surjective setting.
  As our main result, we extend the class EDS to arbitrary finite domains and
provide a conditional complexity classification for surjective VCSPs of this
type based on a reduction to smaller domains. On three-element domains, this
leads to a complete classification of such VCSPs.
