The minimal dominating sets in a directed graph and the key indicators set of socio-economic system

The paper deals with a digraph with non-negative vertex weights. A subset $W$ of the set of vertices is called dominating if any vertex that not belongs to it is reachable from the set $W$ within precisely one step. A dominating set is called minimal if it ceases to be dominating when removing any vertex from it. The paper investigates the problem of searching for a minimal dominating set of maximum weight in a vertex-weighted digraph. An integer linear programming model is proposed for this problem. The model is tested on random instances and the real problem of choosing a family of key indicators in a specific socio-economic system. The paper compares this model with the problem of choosing a dominating set with a fixed number of vertices.