## Taiwanese Journal of Mathematics

### Inverse Anti-$k$-centrum Problem on Networks with Variable Edge Lengths

#### Abstract

This paper concerns the problem of modifying edge lengths of a network at minimum total costs so as to make a prespecified vertex become an optimal location in the modified environment. Here, we focus on the ordered median objective function with respect to the vector of multipliers $\lambda = (1,\ldots,1,0,\ldots,0)$ with $k$ 1's. This problem is called the inverse anti-$k$-centrum problem. We first show that the inverse anti-$k$-centrum problem is NP-hard even on tree networks. However, for the inverse anti-$k$-centrum problem on cycles, we formulate it as one or two linear programs, depending on odd or even integer $k$. Concerning the special cases with $k = 2,3,M$, we develop combinatorial algorithms that efficiently solve the problem, where $M$ is the number of vertices of the cycle.

#### Article information

Source
Taiwanese J. Math., Advance publication (2019), 22 pages.

Dates
First available in Project Euclid: 21 June 2019

https://projecteuclid.org/euclid.twjm/1561082413

Digital Object Identifier
doi:10.11650/tjm/190602

#### Citation

Pham, Van Huy; Nguyen, Kien Trung. Inverse Anti-$k$-centrum Problem on Networks with Variable Edge Lengths. Taiwanese J. Math., advance publication, 21 June 2019. doi:10.11650/tjm/190602. https://projecteuclid.org/euclid.twjm/1561082413

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