Abstract
We consider the problem of modifying edge lengths of a cycle at minimum total costs so as to make a prespecified vertex an (absolute) 1-center of the cycle with respect to the new edge legths. We call this problem the inverse 1-center problem on a cycle. To solve this problem, we first construct the so-called optimality criterion for a vertex to be a 1-center. Based on the optimality criterion, it is shown that the problem can be separated into linearly many subproblems. For a predetermined subproblem, we apply a parameterization approach to formulate it as a minimization problem of a piecewise linear convex function with a connected feasible region. Hence, it is shown that the problem can be solved in \(O(n^2 \log n)\) time, where n is the number of vertices in the cycle.
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The author would like to acknowledge the anonymous referees for their valuable comments, which helped to improve the paper.
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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2016.08.
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Nguyen, K.T. The inverse 1-center problem on cycles with variable edge lengths. Cent Eur J Oper Res 27, 263–274 (2019). https://doi.org/10.1007/s10100-017-0509-4
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DOI: https://doi.org/10.1007/s10100-017-0509-4