Flows Local Control in Resource Networks with A Low Resource
Vladimir Aleksandrovich Skorokhodov, Iakov M. Erusalimskiy · Mathematics and Statistics · 2023
The flow control problem in resource networks consists in finding such a set of vertices and capacities of arcs, which go out from these vertices, such that the limit state of the resource network is the closest to the given state . This problem is naturally divided into two subproblems. The first of them is the ”local” subproblem, which consists in determining the capacities of arcs which go out from the vertices of a given subset (hereinafter, the set will be called the set of controlled vertices). The second subproblem is the ”global” subproblem, which consists in finding the optimal set of controlled vertices , consisting of at most s elements. The paper is devoted to the study of the possibility of flows local control in resource networks. Methods for solving a local subproblem for regular resource networks with a low resource allocation are proposed. The conditions for the unreachability of the limit state , which coincides with the given state are obtained. Three cases are considered for the distribution of controlled vertices on a resource network. In each of the considered cases, it is shown that if the condition of unreachability of the limit state is not satisfied, then there is a set of the capacities values of the arcs that go out the controlled vertices, for which the limit state coincides with the state .