Central European Journal of Operations Research

, Volume 27, Issue 1, pp 263–274

# The inverse 1-center problem on cycles with variable edge lengths

• Kien Trung Nguyen
Original Paper

## 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.

## Keywords

1-Center problem Inverse optimization Cycle Convex Parameterization

## Mathematics Subject Classification

90B10 90B80 90C27

## Notes

### Acknowledgements

The author would like to acknowledge the anonymous referees for their valuable comments, which helped to improve the paper.

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