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The classical reverse 1-median problem on trees is to adjust the edge lengths within a budget so as to reduce the 1-median function at a predetermined vertex as much as possible. This paper concerns the corresponding problem with uncertain vertex weights presented by linear functions. Moreover, we use the minmax regret criterion to measure the maximum loss of a feasible solution with respect to the worst-case scenario. The regarding problem is called the minmax regret reverse 1-median problem on trees. We first partition the set of scenarios into parts such that the optimal solution of the corresponding reverse 1-median problem does not change in each part. Then the problem can be reformulated as the minimization of a quadratic number of affine linear functions. We finally develop a greedy algorithm that solves the problem in $O(n^3)$ time where n is the number of vertices in the underlying tree.

We address in this paper a variant of the scheduling-location (ScheLoc) problem on tree networks with interval edge lengths where the total deviation of the uncertain data cannot exceed a threshold. We further use the minmax regret concept to deal with the corresponding uncertainty. In order to solve the problem, we investigate the structure of the schedule which leads to the maximum regret value at a fixed point. Then we consider the machine location belonging to a specific edge of the tree and partition the underlying edge into regions with linear maximum regret function. Finally, we develop a combinatorial algorithm that solves the minmax regret ScheLoc problem in polynomial time based on a finite dominating set approach.