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.

给定一棵无向树\(T=(V,E)\)和一个值\(\sigma >0\)，每条边\(e\in E\)都有一个前置时间l ( e ) 和一个容量c ( e）。令\(P_{st}\)为连接s和t的唯一路径。在V中从s发送\(\sigma \)个单位数据到\(t\)的传输时间为\(Q(s,t,\sigma )=l(P_{st})+\frac{\sigma }{c(P_{st})}\)，其中\(l(P_{st})=\sum _{e\in P_{st}}l(e)\)和\(c(P_{st}) })=\min _{e\in P_{st}} c(e)\)。顶点（绝对）最快 1 中心问题是确定顶点\(s^*\in V\)（点\(s^*\in T\)，它可以是 V 中的顶点或内部点某些边），其最大传输时间最小。在树T上的逆顶点（绝对）最快 1 中心问题中，我们的目标是在加权\(l_1\)范数下以最小成本修改容量向量，使得给定顶点（点）成为顶点（绝对）最快的 1 中心。我们首先引入两棵树\(T_1\)和\(T_2\)之间的最大传输时间平衡问题，其中我们减少\(T_1\)的最大传输时间并增加\(T_2\)的最大传输时间，直到两棵树的最大传输时间变得相等。我们提出问题的目标函数的解析形式，然后设计一个\(O(n_1^2n_2)\)算法，其中\(n_i\)是\(T_i\)的顶点数，其中\(i= 1, 2\) . 此外，我们分析了两个反问题的一些最优性条件，这支持我们将它们转化为相应的最大传输时间平衡问题。最后，我们提出了两种O(n^3)算法，其中n是T中的顶点数量。