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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.13767 |
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| _version_ | 1866908409437093888 |
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| author | Maji, Sukanya Pandit, Supantha Sadhu, Sanjib |
| author_facet | Maji, Sukanya Pandit, Supantha Sadhu, Sanjib |
| contents | We study the Generalized Red-Blue Annulus Cover problem for two sets of points, red ($R$) and blue ($B$), where each point $p \in R\cup B$ is associated with a positive penalty ${\cal P}(p)$. The red points have non-covering penalties, and the blue points have covering penalties. The objective is to compute an annulus (either a rectangular or a circular) $\cal A$ such that the value of the function ${\cal P}({R}^{out}) + {\cal P}({ B}^{in})$ is minimum, where ${R}^{out} \subseteq {R}$ is the set of red points not covered by ${\cal A}$, and ${B}^{in} \subseteq {B}$ is the set of blue points covered by $\cal A$. We study the problem for various types of axis-parallel rectangular annulus and circular annulus in one and two dimensions. We also study a restricted version of the rectangular annulus cover problem, where the center of the annulus is constrained to lie on a given horizontal line $L$. We design a polynomial-time algorithm for each type of annulus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymmetric Separation Problem for Bichromatic Point Set Maji, Sukanya Pandit, Supantha Sadhu, Sanjib Computational Geometry We study the Generalized Red-Blue Annulus Cover problem for two sets of points, red ($R$) and blue ($B$), where each point $p \in R\cup B$ is associated with a positive penalty ${\cal P}(p)$. The red points have non-covering penalties, and the blue points have covering penalties. The objective is to compute an annulus (either a rectangular or a circular) $\cal A$ such that the value of the function ${\cal P}({R}^{out}) + {\cal P}({ B}^{in})$ is minimum, where ${R}^{out} \subseteq {R}$ is the set of red points not covered by ${\cal A}$, and ${B}^{in} \subseteq {B}$ is the set of blue points covered by $\cal A$. We study the problem for various types of axis-parallel rectangular annulus and circular annulus in one and two dimensions. We also study a restricted version of the rectangular annulus cover problem, where the center of the annulus is constrained to lie on a given horizontal line $L$. We design a polynomial-time algorithm for each type of annulus. |
| title | Asymmetric Separation Problem for Bichromatic Point Set |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2402.13767 |