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

## TL;DR

This paper introduces a polynomial-time algorithm for a generalized Min-Cut problem with bounds and applies it to classify the complexity of surjective VCSPs over finite domains, extending known results from Boolean to larger domains.

## Contribution

It develops a new algorithm for a generalized Min-Cut problem and extends the classification of surjective VCSPs from Boolean to arbitrary finite domains.

## Key findings

- Polynomial-time enumeration of near-optimal solutions for the generalized Min-Cut problem.
- Extension of the EDS tractable class to arbitrary finite domains.
- Complete classification of surjective VCSPs on three-element domains.

## Abstract

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.