The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity (CLP), each level $y$ is equipped with a partial order $\prec_y$ on its vertices and in the desired drawing the left-to-right order of vertices on level $y$ has to be a linear extension of $\prec_y$. Ordered Level Planarity (OLP) corresponds to the special case of CLP where the given partial orders $\prec_y$ are total orders. Previous results by Br\"uckner and Rutter [SODA 2017] and Klemz and Rote [ACM Trans. Alg. 2019] state that both CLP and OLP are NP-hard even in severely restricted cases. In particular, they remain NP-hard even when restricted to instances whose width (the maximum number of vertices that may share a common level) is at most two. In this paper, we focus on the other dimension: we study the parameterized complexity of CLP and OLP with respect to the height (the number of levels). We show that OLP parameterized by the height is complete with respect to the complexity class XNLP, which was first studied by Elberfeld et al. [Algorithmica 2015] (under a different name) and recently made more prominent by Bodlaender et al. [FOCS 2021]. It contains all parameterized problems that can be solved nondeterministically in time $f(k) n^{O(1)}$ and space $f(k) \log n$ (where $f$ is a computable function, $n$ is the input size, and $k$ is the parameter). If a problem is XNLP-complete, it lies in XP, but is W[$t$]-hard for every $t$. In contrast to the fact that OLP parameterized by the height lies in XP, it turns out that CLP is NP-hard even when restricted to instances of height 4. We complement this result by showing that CLP can be solved in polynomial time for instances of height at most 3.

翻译：层级平面性问题要求在图平面中实现无交叉绘制，使得顶点被放置在预设的y坐标（称为层级）上，并且每条边以y单调曲线形式呈现。在约束层级平面性（CLP）变体中，每个层级$y$上的顶点配备一个偏序关系$\prec_y$，在期望的绘图中，层级$y$上顶点的从左到右顺序必须为$\prec_y$的线性扩展。有序层级平面性（OLP）对应于CLP中给定偏序$\prec_y$为全序的特例。Brückner与Rutter [SODA 2017] 及Klemz与Rote [ACM Trans. Alg. 2019] 的先前研究指出，即使在严重受限情形下，CLP与OLP均为NP困难问题。特别地，当实例约束为宽度（共享同一层级的最大顶点数）不超过2时，它们仍保持NP困难性。本文聚焦另一维度：研究CLP与OLP关于高度（层级数量）的参数复杂性。我们证明，以高度为参数的OLP对于复杂度类XNLP是完全的，该复杂度类最初由Elberfeld等人 [Algorithmica 2015] 提出（当时使用不同名称），近期因Bodlaender等人 [FOCS 2021] 的研究而显著化。该类包含所有可在时间$f(k) n^{O(1)}$和空间$f(k) \log n$内非确定性求解的参数化问题（其中$f$为可计算函数，$n$为输入规模，$k$为参数）。若某问题为XNLP完全，则它属于XP类，但对所有$t$均为W[$t$]困难。与以高度为参数的OLP属于XP类的事实相反，我们发现CLP在高度仅为4的实例上仍为NP困难。作为补充，我们证明当实例高度不超过3时，CLP可在多项式时间内求解。