Inverse Vertex/Absolute Quickest 1-Center Location Problem on a Tree Under Weighted l_1 Norm

Given an undirected tree T=(V,E) and a value σ >0 , every edge e∈ E has a lead time l ( e ) and a capacity c ( e ). Let P_st be the unique path connecting s and t . A transmission time of sending σ units data from s to t∈ V is Q(s,t,σ )=l(P_st)+σ/c(P_st) , where l(P_st)=∑ _e∈ P_stl(e) and c(P_st)=min _e∈ P_st c(e) . A vertex (an absolute) quickest 1-center problem is to determine a vertex s^*∈ V (a point s^*∈ T , which is either a vertex or an interior point in some edge) whose maximum transmission time is minimum. In an inverse vertex (absolute) quickest 1-center problem on a tree T , we aim to modify a capacity vector with minimum cost under weighted l_1 norm such that a given vertex (point) becomes a vertex (an absolute) quickest 1-center. We first introduce a maximum transmission time balance problem between two trees T_1 and T_2 , where we reduce the maximum transmission time of T_1 and increase the maximum transmission time of T_2 until the maximum transmission time of the two trees become equal. We present an analytical form of the objective function of the problem and then design an O(n_1^2n_2) algorithm, where n_i is the number of vertices of T_i with i=1, 2 . Furthermore, we analyze some optimality conditions of the two inverse problems, which support us to transform them into corresponding maximum transmission time balance problems. Finally, we propose two O(n^3) algorithms, where n is the number of vertices in T .