In this article, we consider colorable variations of the Unit Disk Cover ({\it UDC}) problem as follows. {\it $k$-Colorable Discrete Unit Disk Cover ({\it $k$-CDUDC})}: Given a set $P$ of $n$ points, and a set $D$ of $m$ unit disks (of radius=1), both lying in the plane, and a parameter $k$, the objective is to compute a set $D'\subseteq D$ such that every point in $P$ is covered by at least one disk in $D'$ and there exists a function $\chi:D'\rightarrow C$ that assigns colors to disks in $D'$ such that for any $d$ and $d'$ in $D'$ if $d\cap d'\neq\emptyset$, then $\chi(d)\neq\chi(d')$, where $C$ denotes a set containing $k$ distinct colors. For the {\it $k$-CDUDC} problem, our proposed algorithms approximate the number of colors used in the coloring if there exists a $k$-colorable cover. We first propose a 4-approximation algorithm in $O(m^{7k}n\log k)$ time for this problem, where $k$ is a positive integer. The previous best known result for the problem when $k=3$ is due to the recent work of Biedl et al. [CCCG 2019], who proposed a 2-approximation algorithm in $O(m^{25}n)$ time. For $k=3$, our algorithm runs in $O(m^{21}n)$ time, faster than the previous best algorithm, but gives a 4-approximate result. We then generalize our approach to yield a family of $\rho$-approximation algorithms in $O(m^{\alpha k}n\log k)$ time, where $(\rho,\alpha)\in \{(4, 7), (6,5), (7, 5), (9,4)\}$. We further generalize this to exhibit a $O(\frac{1}{\tau})$-approximation algorithm in $O(m^{\alpha k}n\log k)$ time for a given grid width $1 \leq \tau \leq 2$, where $\alpha=O(\tau^2)$. We also extend our algorithm to solve the {\it $k$-Colorable Line Segment Disk Cover ({\it $k$-CLSDC})} and {\it $k$-Colorable Rectangular Region Cover ({\it $k$-CRRC})} problems, in which instead of the set $P$ of $n$ points, we are given a set $S$ of $n$ line segments, and a rectangular region $\cal R$, respectively.

中文翻译：

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$ k $-彩色单元磁盘盖问题

在本文中，我们考虑了单位磁盘封面（{\ it UDC}）问题的彩色变化，如下所示。{\ it $ k $ -Colorable Discrete Unit Disk Cover（{\ it $ k $ -Colorable Discrete Unit Disk Cover（{\ it $ k $ -CDUDC}）}：给定一组$ P $的$ n $点和一组$ D $的$ m $单位盘（分别位于平面和参数$ k $的范围内，其目的是计算一个集合$ D'\ subseteq D $，以使$ P $中的每个点至少被$中的一个磁盘覆盖D'$，并且存在一个函数$ \ chi：D'\ rightarrow C $，该函数将颜色分配给$ D'$中的磁盘，这样，对于$ d'和$ D'$中的任何$ d $和$ d'$，如果$ d \ cap d'\ neq \ emptyset $，然后是$ \ chi（d）\ neq \ chi（d'）$，其中$ C $表示包含$ k $种不同颜色的集合。对于{\ it $ k $ -CDUDC}问题，如果存在$ k $可着色的封面，我们提出的算法将近似用于着色的颜色数。我们首先针对这个问题在$ O（m ^ {7k} n \ log k）$时间内提出一种4近似算法，其中$ k $是一个正整数。$ k = 3 $时该问题的先前最著名的结果是由于Biedl等人的最新工作。[CCCG 2019]，他提出了$ O（m ^ {25} n）$时间的2近似算法。对于$ k = 3 $，我们的算法以$ O（m ^ {21} n）$的时间运行，比之前的最佳算法要快，但结果却是4左右。然后，我们一般化我们的方法，以在$ O（m ^ {\ alpha k} n \ log k）$的时间内产生$ \ rho $-近似算法族，其中$ {\ rho，\ alpha）\ in \ {（ 4，7），（6,5），（7，5），（9,4）\} $。对于给定的网格宽度$ 1 \ leq，我们进一步将其推广为在$ O（m ^ {\ alpha k} n \ log k）$时间中表现出$ O（\ frac {1} {\ tau}）$近似算法\ tau \ leq 2 $，其中$ \ alpha = O（\ tau ^ 2）$。