AI 中文总结
研究具有平面（哈斯）图的偏序集维度计算问题，给出常数因子多项式时间近似算法，通过对标准示例的结构理解得出算法结果，证明了\(\mathrm{dim}(P) \leq 96\mathrm{se}(P)+672\)这一更强界。
AI 中文摘要
偏序集\(P\)的维度是最小正整数\(d\)，使得\(P\)是配备乘积序的\(\mathbb{R}^d\)的诱导子偏序集。我们给出了一个常数因子多项式时间近似算法，用于计算具有平面（哈斯）图的偏序集类中的维度。虽然计算偏序集的维度一般是NP难的，但平面偏序集问题的计算复杂性仍然未知。该算法结果源于对小维度的典型障碍——标准示例的结构理解。一个长期存在的问题，始于20世纪80年代初，询问每个具有平面图的偏序集的维度是否由其包含的标准示例的最大阶数的函数界定。在该系列的第一篇论文中，我们通过建立多项式界在更一般的具有平面覆盖图的偏序集设置中解决了该问题。我们在原始设置中证明了一个更强的界，即对于每个具有平面图的偏序集\(P\)，\(\mathrm{dim}(P) \leq 96\mathrm{se}(P)+672\)，其中\(\mathrm{dim}(P)\)表示\(P\)的维度，\(\mathrm{se}(P)\)表示\(P\)中包含的标准示例的最大阶数。
英文摘要
The dimension of a poset $P$ is the minimum positive integer $d$ such that $P$ is an induced subposet of $\mathbb{R}^d$ equipped with the product order. We give a constant-factor polynomial-time approximation algorithm for computing dimension in the class of posets with a planar (Hasse) diagram. While computing the dimension of a poset is NP-hard in general, the computational complexity of the problem for planar posets remains open. The algorithmic result is driven by a structural understanding of the canonical obstruction to small dimension: standard examples. A longstanding problem, originating in the early 1980s, asked whether every poset with a planar diagram has dimension bounded by a function of the maximum order of a standard example that it contains. In the first paper of the series, we have resolved the problem in a more general setting of posets with planar cover graphs by establishing a polynomial bound. We prove a stronger bound in the original setting, namely, for every poset $P$ with a planar diagram $\mathrm{dim}(P) \leq 96\mathrm{se}(P)+672$, where $\mathrm{dim}(P)$ denotes the dimension of $P$ and $\mathrm{se}(P)$ denotes the maximum order of a standard example contained in $P$.
Comments57 pages. arXiv admin note: text overlap with arXiv:2510.18603
