Given an edge-weighted graph and a subset of the vertices called terminals, the multiway cut problem aims to find a minimum weight set of edges that separates each terminal from all the others. This problem is known to be NP-hard even for \(k=3\). Computational experiments on social networks obtained from the Indian e-commerce company Flipkart show that an integer programming formulation (referred to as EF2) of the problem introduced by Chopra and Owen (1996) provides a very strong LP-relaxation that allows us to solve large problems in a reasonable amount of time. We show that the cardinality EF2 (where all edge weights are 1) on a wheel graph has a primal integer solution and a dual integer solution of the same value. We consider a hub-spoke network of wheel graphs constructed by adding edges connecting the hub vertices of a collection of wheel graphs. We assume that every wheel has a terminal hub or a terminal vertex with three non-terminal neighbors, and show that if the graph connecting the hub vertices is planar, the cardinality EF2 on the hub-spoke network of the wheel graphs has a primal integer solution and a dual integer solution of the same value. Given the prevalence of such structures in our social networks, our results provide some theoretical justification for the strong empirical performance of the EF2 formulation.

给定一个边加权图和一个称为终端的顶点子集，多路切割问题旨在找到一个最小权重的边集，将每个终端与所有其他终端分开。即使对于\(k=3\)，这个问题也是已知的 NP-hard. 从印度电子商务公司 Flipkart 获得的社交网络上的计算实验表明，Chopra 和 Owen (1996) 引入的问题的整数规划公式（称为 EF2）提供了非常强大的 LP 松弛，使我们能够解决大型在合理的时间内解决问题。我们证明了轮图上的基数 EF2（其中所有边权重为 1）具有相同值的原始整数解和对偶整数解。我们考虑通过添加连接轮图集合的轮毂顶点的边来构建轮图的轮毂辐条网络。我们假设每个轮子都有一个终端轮毂或一个终端顶点与三个非终端邻居，并证明如果连接轮毂顶点的图形是平面的，轮图的轮辐网络上的基数 EF2 具有相同值的原始整数解和对偶整数解。鉴于此类结构在我们的社交网络中普遍存在，我们的结果为 EF2 公式的强大经验表现提供了一些理论依据。