度量维数及其变体的节流（Throttling）研究
Throttling for metric dimension and its variants
AI总结：
该研究提出度量维数及其边、混合变体的节流方法，证明其节流数计算为NP-hard，给出特殊图族节流数公式与极值图特征，推导了n阶图与n阶树的最小节流数阶数界。
摘要：
度量维数是一种已应用于机器人导航与低维向量嵌入求解的图参数。节流（Throttling）指在求解特定图问题时最小化两种可用资源之和的操作。本文提出了度量维数、边度量维数与混合度量维数的节流方法。在向量嵌入场景下，度量维数节流可生成具有整数坐标的低维、低幅值嵌入。本文证明三类变体的节流数计算均为NP-hard问题，给出了特殊图族的节流数计算公式，并刻画了具有极值节流数的图的特征。此外还证明，n阶图的最小可能节流数为$\Theta\left(\frac{\log{n}}{\log{\log{n}}}\right)$，而n阶树的最小可能节流数根据度量维数变体的不同，为$\Theta(n^{1/3})$或$\Theta(n^{1/2})$。
英文摘要：
Metric dimension is a graph parameter that has been applied to robot navigation and finding low-dimensional vector embeddings. Throttling entails minimizing the sum of two available resources when solving certain graph problems. In this paper, we introduce throttling for metric dimension, edge metric dimension, and mixed metric dimension. In the context of vector embeddings, metric dimension throttling finds a low-dimensional, low-magnitude embedding with integer coordinates. We show that computing the throttling number is NP-hard for all three variants. We give formulas for the throttling numbers of special families of graphs, and characterize graphs with extremal throttling numbers. We also prove that the minimum possible throttling number of a graph of order $n$ is $Θ\left(\frac{\log{n}}{\log{\log{n}}}\right)$, while the minimum possible throttling number of a tree of order $n$ is $Θ(n^{1/3})$ or $Θ(n^{1/2})$ depending on the variant of metric dimension.
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