The inverse 1-center problem on cycles with variable edge lengths
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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.
Keywords1-Center problem Inverse optimization Cycle Convex Parameterization
Mathematics Subject Classification90B10 90B80 90C27
The author would like to acknowledge the anonymous referees for their valuable comments, which helped to improve the paper.
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