Location theory on networks is concerned with the problem of selecting the best location in a specified network for facilities. Many studies for the theory have been done. We have studied location theory from the standpoint of measuring the closeness between two vertices by the capacity (maximum flow value) between two vertices. In a previous paper, we have considered location problems, called covering problems and proposed polynomial time algorithms for these problems. These problems are applicable to assigning files to some computers in a computer network. This paper is concerned with a covering problem called the single cover problem defined in the previous paper. First, we define a generalized single cover problem and show that an algorithm proposed in the previous paper can be applicable to solving the generalized single cover problem. Then, we define a single cover problem satisfying cardinality constrains and show that the problem is solved in a polynomial time.

Location theory on networks is concerned with the problem of selecting the best location in a specified network for facilities. Many studies for the theory have been done. However, few studies treat location problems on networks from the standpoint of measuring the closeness between two vertices by the capacity (maximum flow value) between two vertices. This paper concerns location problems, called covering problems on flow networks. We define two types of covering problems on flow networks. We show that covering problems on undirected flow networks and a covering problem on directed flow networks are solved in polynomial times.