We consider the following capacitated covering problem. We are given a set P of n points and a set $${\mathcal {B}}$$ B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in $${\mathcal {B}}$$ B . We would like to compute a subset $${\mathcal {B}}' \subseteq {\mathcal {B}}$$ B ′ ⊆ B of balls and assign each point in P to some ball in $${\mathcal {B}}'$$ B ′ that contains it, so that the number of points assigned to any ball is at most U . The objective function that we would like to minimize is the cardinality of $${\mathcal {B}}'$$ B ′ . We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining “bi-criteria” approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for this problem. In fact, in the Euclidean setting, only $$1+\epsilon $$ 1 + ϵ factor expansion is sufficient for any $$\epsilon > 0$$ ϵ > 0 , with the approximation factor being a polynomial in $$1/\epsilon $$ 1 / ϵ . We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems.

中文翻译：

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几何空间中的电容覆盖问题

我们考虑以下容量覆盖问题。我们给定了一个由 n 个点组成的集合 P 和一个来自某个度量空间的球的集合 $${\mathcal {B}}$$ B，以及一个表示 $${\ 中每个球的容量的正整数 U数学 {B}}$$ B 。我们想计算一个子集 $${\mathcal {B}}' \subseteq {\mathcal {B}}$$ B ′ ⊆ B 球并将 P 中的每个点分配给 $${\mathcal { B}}'$$ B ′ 包含它，因此分配给任何球的点数最多为 U 。我们想要最小化的目标函数是 $${\mathcal {B}}'$$ B ′ 的基数。我们在任意度量空间以及恒定维数的欧几里得空间中考虑这个问题。在度量设置中，即使是无能力版本的问题也很难在对数因子内近似。在欧几里德设置中，3 维及更高维中最著名的近似保证是点数的对数。因此，我们专注于获得“双标准”近似值。特别是，我们可以通过某种因素扩大我们解决方案中的球，但最佳解决方案没有这种灵活性。我们的主要结果是允许输入球的常数因子扩展足以获得该问题的常数近似值。事实上，在欧几里德设置中，对于任何 $$\epsilon > 0$$ ϵ > 0 ，只有 $$1+\epsilon $$ 1 + ϵ 因子扩展就足够了，近似因子是 $$1/\epsilon 中的多项式$$ 1 / ϵ . 我们使用统一的方案来舍入自然 LP 松弛，从而获得这些结果；该方案可能对其他电容覆盖问题有用。