Given an undirected tree \(T=(V,E)\) and a value \(\sigma >0\) , every edge \(e\in E\) 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 \(\sigma \) units data from s to \(t\in V\) is \(Q(s,t,\sigma )=l(P_{st})+\frac{\sigma }{c(P_{st})}\) , where \(l(P_{st})=\sum _{e\in P_{st}}l(e)\) and \(c(P_{st})=\min _{e\in P_{st}} c(e)\) . A vertex (an absolute) quickest 1-center problem is to determine a vertex \(s^*\in V\) (a point \(s^*\in T\) , 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.
