Taiwanese Journal of Mathematics

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

Van Huy Pham and Kien Trung Nguyen

Full-text: Open access

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., Volume 24, Number 2 (2020), 501-522.

Dates
Received: 4 December 2018
Revised: 30 May 2019
Accepted: 13 June 2019
First available in Project Euclid: 21 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1561082413

Digital Object Identifier
doi:10.11650/tjm/190602

Mathematical Reviews number (MathSciNet)
MR4078208

Zentralblatt MATH identifier
07192945

Subjects
Primary: 90B10: Network models, deterministic 90B80: Discrete location and assignment [See also 90C10] 90C27: Combinatorial optimization

Keywords
location problems inverse optimization problems ordered median function anti-$k$-centrum tree cycle

Citation

Pham, Van Huy; Nguyen, Kien Trung. Inverse Anti-$k$-centrum Problem on Networks with Variable Edge Lengths. Taiwanese J. Math. 24 (2020), no. 2, 501--522. doi:10.11650/tjm/190602. https://projecteuclid.org/euclid.twjm/1561082413


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