舍入的几何：近似最优界与一个新的邻域 Sperner 引理
Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma
AI总结：
本文研究了$(k,\varepsilon)$-隔离划分，建立了$k$与$\varepsilon$之间的近似最优界，提出了广泛参数范围的新构造，并基于下界推导出超立方体上的新邻域Sperner引理及舍入方案的无免费午餐定理。
摘要：
$\mathbb{R}^d$ 的一个划分 $\mathcal{P}$ 被称为 $(k,\varepsilon)$-隔离划分，如果对于每个 $\vec{p} \in \mathbb{R}^d$，球 $\overline{B}_{\infty}(\varepsilon, \vec{p})$ 最多与 $\mathcal{P}$ 的 $k$ 个成员相交。设计此类隔离划分的目标是最小化 $k$ 同时使 $\varepsilon$ 尽可能大。该划分问题与多个主题相关，包括确定性舍入方案、伪确定性、可复制性以及 Sperner/KKM 型结果。在本工作中，我们建立了 $k$ 和 $\varepsilon$ 之间的近似最优关系。我们证明，对于任何有界测度划分和任何 $d\geq 1$，必然有 $k\geq(1+2\varepsilon)^d$。因此，当 $k=k(d)$ 被限制为 ${\rm poly}(d)$ 时，可得 $\varepsilon=\varepsilon(d)\in O\left(\frac{\ln d}{d}\right)$。该界在对数因子范围内是紧的，因为已知存在 $k(d)=d+1$ 和 $\varepsilon(d)=\frac{1}{2d}$ 的隔离划分。我们还提供了适用于广泛 $k(d)$ 和 $\varepsilon(d)$ 参数范围的隔离划分的新构造。具体而言，我们证明对于任何 $f:\mathbb{N}\rightarrow\mathbb{N}$，存在一个 $k(d)=(f(d)+1)^{\lceil\frac{d}{f(d)}\rceil}$ 且 $\varepsilon(d)=\frac{1}{2f(d)}$ 的隔离划分。对于 $k(d)$ 和 $\varepsilon(d)$ 的各种选择，这些新划分在最优化程度上仅差 $O(\log d)$ 因子。基于下界结果，我们在超立方体上建立了一个新的邻域版本的 Sperner 引理，该引理具有独立的研究价值。此外，我们证明了一个关于伪确定性/可复制算法背景下舍入方案局限性的无免费午餐定理。
英文摘要：
A partition $\mathcal{P}$ of $\mathbb{R}^d$ is called a $(k,\varepsilon)$-secluded partition if, for every $\vec{p} \in \mathbb{R}^d$, the ball $\overline{B}_{\infty}(\varepsilon, \vec{p})$ intersects at most $k$ members of $\mathcal{P}$. A goal in designing such secluded partitions is to minimize $k$ while making $\varepsilon$ as large as possible. This partition problem has connections to a diverse range of topics, including deterministic rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type results. In this work, we establish near-optimal relationships between $k$ and $\varepsilon$. We show that, for any bounded measure partitions and for any $d\geq 1$, it must be that $k\geq(1+2\varepsilon)^d$. Thus, when $k=k(d)$ is restricted to ${\rm poly}(d)$, it follows that $\varepsilon=\varepsilon(d)\in O\left(\frac{\ln d}{d}\right)$. This bound is tight up to log factors, as it is known that there exist secluded partitions with $k(d)=d+1$ and $\varepsilon(d)=\frac{1}{2d}$. We also provide new constructions of secluded partitions that work for a broad spectrum of $k(d)$ and $\varepsilon(d)$ parameters. Specifically, we prove that, for any $f:\mathbb{N}\rightarrow\mathbb{N}$, there is a secluded partition with $k(d)=(f(d)+1)^{\lceil\frac{d}{f(d)}\rceil}$ and $\varepsilon(d)=\frac{1}{2f(d)}$. These new partitions are optimal up to $O(\log d)$ factors for various choices of $k(d)$ and $\varepsilon(d)$. Based on the lower bound result, we establish a new neighborhood version of Sperner's lemma over hypercubes, which is of independent interest. In addition, we prove a no-free-lunch theorem about the limitations of rounding schemes in the context of pseudodeterministic/replicable algorithms.
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