In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.

中文翻译：

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单位圆盘图上的广义独立支配集问题

在本文中，我们研究了最大独立集和最小支配集问题的广义版本，即单位圆盘图上的最大 $d$-距离独立集问题和最小 $d$-距离支配集问题整数 $d>0$。我们首先证明最大$d$-distance 独立集问题和最小$d$-distance 支配集问题属于NP-hard 类。接下来，我们针对这两个问题提出了一个简单的多项式时间常数因子近似算法和 PTAS。