In location theory, group median generalizes the concepts of both median and center. We address in this paper the problem of modifying vertex weights of a tree at minimum total cost so that a prespecified vertex becomes a group 1-median with respect to the new weights. We call this problem the inverse group 1-median on trees. To solve the problem, we first reformulate the optimality criterion for a vertex being a group 1-median of the tree. Based on this result, we prove that the problem is $ NP $-hard. Particularly, the corresponding problem with exactly two groups is however solvable in $ O(n^2\log n) $ time, where $ n $ is the number of vertices in the tree.

中文翻译：

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树上的第1组中值反问题

在位置理论中，组中位数概括了中位数和中心的概念。我们在本文中解决了以最小的总成本修改树的顶点权重的问题，以使预先指定的顶点相对于新权重成为1个中位数。我们称此问题为树上的1中值逆组。为了解决该问题，我们首先为树的第1组中值的顶点重新制定最优准则。根据此结果，我们证明问题是$ NP $-困难的。特别地，然而，恰好具有两个组的对应问题可以在$ O（n ^ 2 \ log n）$时间内解决，其中$ n $是树中顶点的数量。