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

Abstract

This paper studies the continuous connected 2-facility location problem (CC2FLP) in trees. Let $T = (V, E, c, d, \ell , \mu )$ be an undirected rooted tree, where each node $v \in V$ has a weight $d(v) \ge 0$ denoting the demand amount of v as well as a weight $\ell (v) \ge 0$ denoting the cost of opening a facility at v, and each edge $e \in E$ has a weight $c(e) \ge 0$ denoting the cost on e and is associated with a function $\mu (e,t) \ge 0$ 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 $\mathcal {D} \subseteq V$ of clients, and a subset $\mathcal {F} \subseteq \mathcal {P}(T)$ of continuum points admitting facilities where $\mathcal {P}(T)$ is the set of all the points on edges of T, when two facilities are installed at a pair of continuum points $x_1$ and $x_2$ in $\mathcal {F}$, the total cost involved in CC2FLP includes three parts: the cost of opening two facilities at $x_1$ and $x_2$, K times the cost of connecting $x_1$ and $x_2$, and the cost of all the clients in $\mathcal {D}$ connecting to some facility. The objective is to open two facilities at a pair of continuum points in $\mathcal {F}$ to minimize the total cost, for a given input parameter $K \ge 1$. This paper focuses on the case of $\mathcal {D} = V$ and $\mathcal {F} = \mathcal {P}(T)$. 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 $\mathcal {D} \subseteq V$ and $\mathcal {F} \subseteq \mathcal {P}(T)$.