P-curve: An improved solution for the single-facility location problem in two regions with ℓp- and ℓq-norms
Abstract
The distance between two points is provided by the length of their test connecting path. However, there is no unique way to attain this path in the space R2, which is split by a straight line L into two regions, Ωp and Ωq, with ℓp- and ℓq-norms, respectively, where $1<p\leq q $ and L⊂Ωq. In this paper, given a point P∈ Ωp, the locus of points Q∈Ωp, called p-curve, is found, such that there are two ways to attain the distance between P and Q. For each P∈ Ωp, there are two p-curves which are two branches of two different p-parabola. These two branches split Ωp into three subregions using a three-link shortest path in two of these regions. The implicit equation of each p-parabola is given. Furthermore, by using the p-curve, an improved algorithm, called P-MFP, based on the MFP algorithm, is developed in order to solve the single-facility location problem in two regions with ℓp and ℓq-norms. The results obtained with the new algorithm are better than those obtained with the MFP algorithm.ike subheadings, citations, or equations are permitted.
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BibTeX
@misc{franco2026pcurve,
title = {{P-curve: An improved solution for the single-facility location problem in two regions with $\ell_p$- and $\ell_q$-norms}},
author = {Luis Franco and Francisco Velasco and Francisco J. Ortega-Irizo and Luis Gonzalez-Abril},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01340},
eprint = {2610.01340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01340},
}
RIS
TY - UNPB
TI - P-curve: An improved solution for the single-facility location problem in two regions with ℓ_p- and ℓ_q-norms
AU - Luis Franco
AU - Francisco Velasco
AU - Francisco J. Ortega-Irizo
AU - Luis Gonzalez-Abril
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01340
UR - https://arxiv.org/abs/2610.01340
AB - The distance between two points is provided by the length of their test connecting path. However, there is no unique way to attain this path in the space R^2, which is split by a straight line L into two regions, Ω_p and Ω_q, with ℓ_p- and ℓ_q-norms, respectively, where $1<p\leq q $ and L⊂Ω_q. In this paper, given a point P∈ Ω_p, the locus of points Q∈Ω_p, called p-curve, is found, such that there are two ways to attain the distance between P and Q. For each P∈ Ω_p, there are two p-curves which are two branches of two different p-parabola. These two branches split Ω_p into three subregions using a three-link shortest path in two of these regions. The implicit equation of each p-parabola is given. Furthermore, by using the p-curve, an improved algorithm, called P-MFP, based on the MFP algorithm, is developed in order to solve the single-facility location problem in two regions with ℓ_p and ℓ_q-norms. The results obtained with the new algorithm are better than those obtained with the MFP algorithm.ike subheadings, citations, or equations are permitted.
ER -