论文标题
部分可观测时空混沌系统的无模型预测
Covering and packing with homothets of limited capacity
论文作者
论文摘要
这项工作围绕以下两个问题围绕：给定凸件$ c \ subset \ mathbb {r}^d $，一个正整数$ k $和有限的套装$ s \ subset \ subset \ mathbb {r}^d $（或有限的borel borel measure y Mathbb in $ \ mathbb的$ \ r} $ {允许Hyothet覆盖$ S $的$ K $点（或大于$ K $）？如果每个$ c $的同型$ c $都必须覆盖$ s $（或至少有$ k $）的至少$ k $点？我们证明，只要$ s $不太退化，两个问题的答案是$θ_d（\ frac {| s |} {k} {k}）$，其中隐藏的常数独立于$ d $。这是最佳的乘法常数。在措施的情况下，类似的结果成立。然后，我们介绍了凸面$ c $的标准覆盖和包装密度的概括，以$ \ mathbb {r}^d $中的Borel测量空间，并使用上述范围，我们表明它们分别通过$ d $的函数从上方和下方界限。作为中间结果，我们简单地证明了$ o（\ frac {1}ε）的弱$ε$ - $（\ frac {1}ε）的存在，用于$ c $的所有hypothets诱导的范围空间。在最近的一些离散几何形状方面的一些工作之后，我们更详细地调查了$ d = k = 2 $的情况。我们还提供了多项式时间算法，用于构建展示$θ_d的包装/覆盖（\ frac {| s |} {k} {k} {k}）$上面在上面提到的，因为$ c $是欧几里得球。最后，结果表明，如果$ c $是一个正方形，那么决定是否可以使用$ \ frac {| s |} {4} $正方形来决定是否可以覆盖$ s $，每个$ s $每个。
This work revolves around the two following questions: Given a convex body $C\subset\mathbb{R}^d$, a positive integer $k$ and a finite set $S\subset\mathbb{R}^d$ (or a finite Borel measure $μ$ on $\mathbb{R}^d$), how many homothets of $C$ are required to cover $S$ if no homothet is allowed to cover more than $k$ points of $S$ (or have measure larger than $k$)? How many homothets of $C$ can be packed if each of them must cover at least $k$ points of $S$ (or have measure at least $k$)? We prove that, so long as $S$ is not too degenerate, the answer to both questions is $Θ_d(\frac{|S|}{k})$, where the hidden constant is independent of $d$. This is optimal up to a multiplicative constant. Analogous results hold in the case of measures. Then we introduce a generalization of the standard covering and packing densities of a convex body $C$ to Borel measure spaces in $\mathbb{R}^d$ and, using the aforementioned bounds, we show that they are bounded from above and below, respectively, by functions of $d$. As an intermediate result, we give a simple proof the existence of weak $ε$-nets of size $O(\frac{1}ε)$ for the range space induced by all homothets of $C$. Following some recent work in discrete geometry, we investigate the case $d=k=2$ in greater detail. We also provide polynomial time algorithms for constructing a packing/covering exhibiting the $Θ_d(\frac{|S|}{k})$ bound mentioned above in the case that $C$ is an Euclidean ball. Finally, it is shown that if $C$ is a square then it is NP-hard to decide whether $S$ can be covered using $\frac{|S|}{4}$ squares containing $4$ points each.
