FPT Constant Approximation Algorithms for Colorful Sum of Radii
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arXiv
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| Format: | Preprint |
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2025
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| author | Liu, Shuilian Gutin, Gregory Xu, Yicheng Zhang, Yong |
| author_facet | Liu, Shuilian Gutin, Gregory Xu, Yicheng Zhang, Yong |
| contents | We study the colorful sum of radii problem, where the input is a point set $P$ partitioned into classes $P_1, P_2, \dots, P_ω$, along with per-class outlier bounds $m_1, m_2, \dots, m_ω$, summing to $m$. The goal is to select a subset $\mathcal{C} \subseteq P$ of $k$ centers and assign points to centers in $\mathcal{C}$, allowing up to $m_i$ unassigned points (outliers) from each class $P_i$, while minimizing the sum of cluster radii. The radius of a cluster is defined as the maximum distance from any point in the cluster to its center. The classical (non-colorful) version of the sum of radii problem is known to be NP-hard, even on weighted planar graphs. The colorful sum of radii is introduced by Chekuri et al. (2022), who provide an $O(\log ω)$-approximation algorithm. In this paper, we present the first constant-factor approximation algorithms for the colorful sum of radii running in FPT (fixed-parameter tractable) time. Our contributions are twofold: We design an iterative covering algorithm that achieves a $(2+\varepsilon)$-approximation with running time exponential in both $k$ and $m$; We further develop a $(7+\varepsilon)$-approximation algorithm by leveraging a colorful $k$-center subroutine, improving the running time by removing the exponential dependency on $m$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_13191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | FPT Constant Approximation Algorithms for Colorful Sum of Radii Liu, Shuilian Gutin, Gregory Xu, Yicheng Zhang, Yong Computational Geometry Data Structures and Algorithms We study the colorful sum of radii problem, where the input is a point set $P$ partitioned into classes $P_1, P_2, \dots, P_ω$, along with per-class outlier bounds $m_1, m_2, \dots, m_ω$, summing to $m$. The goal is to select a subset $\mathcal{C} \subseteq P$ of $k$ centers and assign points to centers in $\mathcal{C}$, allowing up to $m_i$ unassigned points (outliers) from each class $P_i$, while minimizing the sum of cluster radii. The radius of a cluster is defined as the maximum distance from any point in the cluster to its center. The classical (non-colorful) version of the sum of radii problem is known to be NP-hard, even on weighted planar graphs. The colorful sum of radii is introduced by Chekuri et al. (2022), who provide an $O(\log ω)$-approximation algorithm. In this paper, we present the first constant-factor approximation algorithms for the colorful sum of radii running in FPT (fixed-parameter tractable) time. Our contributions are twofold: We design an iterative covering algorithm that achieves a $(2+\varepsilon)$-approximation with running time exponential in both $k$ and $m$; We further develop a $(7+\varepsilon)$-approximation algorithm by leveraging a colorful $k$-center subroutine, improving the running time by removing the exponential dependency on $m$. |
| title | FPT Constant Approximation Algorithms for Colorful Sum of Radii |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2506.13191 |