A quadratic time exact algorithm for continuous connected 2-facility location problem in trees

This paper studies the continuous connected 2-facility location problem (CC2FLP) in trees. Let be an undirected rooted tree, where each node has a weight denoting the demand amount of v as well as a weight denoting the cost of opening a facility at v, and each edge has a weight denoting the cost on e and is associated with a function denoting the cost of opening a facility at a point x(e, t) on e where t is a continuous variable on e. Given a subset of clients, and a subset of continuum points admitting facilities where is the set of all the points on edges of T, when two facilities are installed at a pair of continuum points and in , the total cost involved in CC2FLP includes three parts: the cost of opening two facilities at and , K times the cost of connecting and , and the cost of all the clients in connecting to some facility. The objective is to open two facilities at a pair of continuum points in to minimize the total cost, for a given input parameter . This paper focuses on the case of and . We first study the discrete version of CC2FLP, named the discrete connected 2-facility location problem (DC2FLP), where two facilities are restricted to the nodes of T, and devise a quadratic time edge-splitting algorithm for DC2FLP. Furthermore, we prove that CC2FLP is almost equivalent to DC2FLP in trees, and develop a quadratic time exact algorithm based on the edge-splitting algorithm. Finally, we adapt our algorithms to the general case of D subset of V and F subset of P(T).