On overlapping territories satisfying cardinality constraints

Given a network with k specified vertices bi called centers, a cardinality constrained cover is a family {Bi} of k subsets covering the vertex set of a network, such that each subset Bi corresponds to and contains center bi, and satisfies a given cardinality constraint. A set of cardinality constrained overlapping territories is a cardinality constrained cover such that the total sum of T(B$_{i}$) for all subsets is minimum among all cardinality constrained covers, where T(B$_{i}$) is the summation of the shortest path lengths from center bi to every vertex in B$_{I}$. This paper considers a problem of finding a set of cardinality constrained overlapping territories. and proposes an algorithm for the Problem which has the time and space complexities are O(k$^{3}$$\mid$V$\mid$$^{2}$) and O(k$\mid$V$\mid$+$\mid$E$\mid$), respectively, where V and E are the sets of vertices and edges of a given network, respectively. The concept of overlapping territories has a possibility to be applied to a job assignment problem.oblem.