A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem

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