Minimizing the sum of distances to a server in a constraint network

This paper deals with constructing centralized geometric network that represents a set of clients that need to be connected to a server. We consider the problem of minimizing the sum of the shortest paths from the clients to the server in the network, subject to some constraints. We refer to this summation as the network cost. The motivation for this work was derived from the combination of centralized communication network and geometric sink spanner. Formally, given a set V of points (representing clients) in the plane, a special point q∈V (representing a server), a real number w>0 (the network weight bound), and an integer h≥1 (the hop bound), the goal is to construct a directed network G=(V,E) of minimum cost, such that there is a directed path from every point in V to q with at most h hops, and ∑ _(v,u)∈E |vu|≤w, where |vu| denotes the Euclidean distance between v and u. In this paper we start by establishing a connection between the considered problem and geometric sink spanner network. Then, we present our main result, a bi-criteria approximation algorithm, that approximates both the weight and the cost of the network with respect to an optimal network, in (|V|/ε) ^O(h/ε) . More precisely, we construct a network such that its cost (i.e., sum of the shortest paths from every v∈V to q) is at most (1+hε) times the cost of an optimal network and its weight (i.e., ∑ _(v,u)∈E |vu|) is at most (1+ε)⋅w, for any ε>0.