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arXiv · 2607.09294

Planarity and dimension II

Heather S. Blake·Jędrzej Hodor·Piotr Micek·Michał T. Seweryn·William T. Trotter
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Abstract

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 $P$.

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Heather S. Blake, Jędrzej Hodor, Piotr Micek, Michał T. Seweryn, William T. Trotter. 2026-07-10. Planarity and dimension II. https://arxiv.org/abs/2607.09294

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