$ j $ | 1 | 2 | 3 | 5 |
$ \bar{q}^1_j $ | 0 | 0 | 0 | 0 |
$ \bar{q}^2_j $ | 0 | 0 | 0 | - |
$ \bar{p}^1_j $ | 0 | 0 | 0 | 3 |
$ \bar{p}^2_j $ | 0 | 0 | 16 | - |
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.
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Table 1. An instance of the inverse 2-group 1-median problem
$ j $ | 1 | 2 | 3 | 5 |
$ \bar{q}^1_j $ | 0 | 0 | 0 | 0 |
$ \bar{q}^2_j $ | 0 | 0 | 0 | - |
$ \bar{p}^1_j $ | 0 | 0 | 0 | 3 |
$ \bar{p}^2_j $ | 0 | 0 | 16 | - |
Table 2.
An instance of the subproblem
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Table 3. Cost values at breakpoints
$ t $ | 8 | 10 | 18 | 20 |
$ C(t) $ | 46 | 44 | 49 | 56 |
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An instance of the inverse 2-group 1-median problem on trees