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 Živný (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.

中文翻译：

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使用 Min-Cut 泛化超越布尔满射 VCSP

在这项工作中，我们首先研究了 Min-Cut 问题的自然泛化，其中一个图通过在其顶点子集上定义的超可加集函数来增强。目标是选择一个顶点子集，使得诱导切割的权重加上设置的函数值最小。此外，对解决方案大小施加了下限和上限。我们提出了一个多项式时间算法，用于枚举这个有界广义最小割问题的所有近似最优解。其次，我们将这种新颖的算法应用于满射广义值约束满足问题（VCSP），即每个标签必须至少使用一次的 VCSP。在布尔域上，Fulla、Uppman 和 Živný (ACM ToCT'18) 最近基于广义最小割问题的无界版本建立了一个完整的满射 VCSP 分类。他们的结果的特点是发现了一个新的非平凡易处理的案例，称为 EDS，它不会出现在非满射设置中。作为我们的主要结果，我们将 EDS 类扩展到任意有限域，并基于减少到更小的域为这种类型的满射 VCSP 提供条件复杂性分类。在三元素域上，这导致了此类 VCSP 的完整分类。