A Polynomial-Time Exact Algorithm for the Connected K-Facility Location Problem on Trees

This paper studies the Connected k-Facility Location Problem (ConkFLP) on trees. Let T = ( V, E, w, rho, f ) be an undirected tree, where V = {v(1), v(2), . . . , v(n)} is the n-vertices set and E = {e(1), e(2), . . . , e(n- 1)} is the (n - 1)-edges set. A facility set F subset of V and a client set C subset of V are given. Each client vi is an element of C has one weight rho(i) >= 0 denoting the demand amount of v(i), and each facility v(i) is an element of F has a weight f(i) >= 0 denoting the opening cost at v(i), and each edge e(j), for j is an element of {1, 2, ... , n - 1}, is associated with a weight w(j) >= 0 denoting the connection cost of it. When some facilities S subset of F are opened, the overall cost involved in ConkFLP includes three parts: the cost of opening facilities S, K times the cost of Steiner tree interconnecting all the opened facilities where K is a fixed parameter, and the total connection cost of assigning each client to the closest facility in S. The goal of ConkFLP is to open at most k < n facilities to minimize the overall cost, for a given input parameter K >= 1. This paper focuses on the case of ConkFLP on trees where F = C = V , and as a result presents a polynomial-time exact dynamic programming algorithm and a computational experiment to illustrate it. Furthermore, a simple way is shown to adapt the algorithm to the general case of F subset of V and C subset of V. Finally, we apply the algorithm to a ConkFLP instance in a regional tree-like water transportation network.