The minimum $s$-$t$ cut problem in graphs is one of the most fundamental problems in combinatorial optimization, and graph cuts underlie algorithms throughout discrete mathematics, theoretical computer science, operations research, and data science. While graphs are a standard model for pairwise relationships, hypergraphs provide the flexibility to model multi-way relationships, and are now a standard model for complex data and systems. However, when generalizing from graphs to hypergraphs, the notion of a "cut hyperedge" is less clear, as a hyperedge's nodes can be split in several ways. Here, we develop a framework for hypergraph cuts by considering the problem of separating two terminal nodes in a hypergraph in a way that minimizes a sum of penalties at split hyperedges. In our setup, different ways of splitting the same hyperedge have different penalties, and the penalty is encoded by what we call a splitting function. Our framework opens a rich space on the foundations of hypergraph cuts. We first identify a natural class of cardinality-based hyperedge splitting functions that depend only on the number of nodes on each side of the split. In this case, we show that the general hypergraph $s$-$t$ cut problem can be reduced to a tractable graph $s$-$t$ cut problem if and only if the splitting functions are submodular. We also identify a wide regime of non-submodular splitting functions for which the problem is NP-hard. We also analyze extensions to multiway cuts with at least three terminal nodes and identify a natural class of splitting functions for which the problem can be reduced in an approximation-preserving way to the node-weighted multiway cut problem in graphs, again subject to a submodularity property. Finally, we outline several open questions on general hypergraph cut problems.

中文翻译：

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具有一般分割函数的超图切割

图中的最小 $s$-$t$ 割问题是组合优化中最基本的问题之一，图割是整个离散数学、理论计算机科学、运筹学和数据科学算法的基础。虽然图是成对关系的标准模型，但超图提供了对多向关系建模的灵活性，并且现在是复杂数据和系统的标准模型。然而，当从图推广到超图时，“切割超边”的概念不太清楚，因为超边的节点可以通过多种方式分割。在这里，我们通过考虑以最小化分裂超边处的惩罚总和的方式分离超图中的两个终端节点的问题来开发超图切割框架。在我们的设置中，分裂同一个超边的不同方式有不同的惩罚，惩罚是由我们所说的分裂函数编码的。我们的框架在超图切割的基础上开辟了丰富的空间。我们首先确定一类自然的基于基数的超边分裂函数，它仅依赖于分裂每一侧的节点数。在这种情况下，我们表明，当且仅当分裂函数是子模时，一般超图 $s$-$t$ 切割问题可以简化为易处理的图 $s$-$t$ 切割问题。我们还确定了问题是 NP-hard 的非子模分裂函数的广泛范围。我们还分析了具有至少三个终端节点的多路切割的扩展，并确定了一类自然的分裂函数，其问题可以以近似保留的方式减少到图中节点加权的多路切割问题，同样受子模块的影响财产。最后，我们概述了关于一般超图切割问题的几个开放性问题。