The minimum cut problem, MC, and the special case of the vertex separator problem, consists in partitioning the set of nodes of a graph G into k subsets of given sizes in order to minimize the number of edges cut after removing the k-th set. Previous work on approximate solutions uses, in increasing strength and expense: eigenvalue, semidefinite programming, SDP, and doubly nonnegative, DNN, bounding techniques. In this paper, we derive strengthened SDP and DNN relaxations, and we propose a scalable algorithmic approach for efficiently evaluating, theoretically verifiable, both upper and lower bounds. Our stronger relaxations are based on a new gangster set, and we demonstrate how facial reduction, FR, fits in well to allow for regularized relaxations. Moreover, the FR appears to be perfectly well suited for a natural splitting of variables, and thus for the application of splitting methods. Here, we adopt the strictly contractive Peaceman-Rachford splitting method, sPRSM. Further, we bring useful redundant constraints back into the subproblems, and show empirically that this accelerates sPRSM.In addition, we employ new strategies for obtaining lower bounds and upper bounds of the optimal value of MC from approximate iterates of the sPRSM thus aiding in early termination of the algorithm. We compare our approach with others in the literature on random datasets and vertex separator problems. This illustrates the efficiency and robustness of our proposed method.

最小割问题，MC，和顶点隔膜问题的特殊情况下，在由划分一个集合图的节点的ģ成ķ给定大小的子集，以便最小化边缘的数量除去后切割ķ第一套。以前关于近似解的工作使用来增加强度和费用：特征值，半定规划SDP和双重非负DNN边界技术。在本文中，我们得出了增强的SDP 和DNN 放松，我们提出了一种可扩展的算法方法，可以有效地评估理论上可验证的上下限。我们更强的松弛是基于一个新的歹徒集，我们演示了如何面部降低，FR，在良好配合，以允许正则放宽。此外，FR 似乎非常适合自然地对变量进行拆分，因此非常适合拆分方法的应用。在这里，我们采用严格收缩的Peaceman-Rachford分裂方法sPRSM。此外，我们带来了有用的冗余约束返回到子问题中，并通过经验证明这可以加速sPRSM。此外，我们采用了新的策略来 从sPRSM的近似迭代中获得MC最佳值的下界和上限， 从而有助于算法的早期终止。我们将我们的方法与有关随机数据集和顶点分隔符问题的文献中的其他方法进行比较。这说明了我们提出的方法的效率和鲁棒性。