Using a min-cut generalisation to go beyond Boolean surjective VCSPs

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.