Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.13702

Constrained and Ordered Level Planarity Parameterized by the Number of Levels

Václav Blažej·Boris Klemz·Felix Klesen·Marie Diana Sieper·Alexander Wolff·Johannes Zink
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 Brückner 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.

Explore related subjects

cs.CG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Václav Blažej, Boris Klemz, Felix Klesen, Marie Diana Sieper, Alexander Wolff, Johannes Zink. 2024-10-19. Constrained and Ordered Level Planarity Parameterized by the Number of Levels. https://arxiv.org/abs/2403.13702

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A constant-factor approximation of the Gromov-Hausdorff distance in the plane

We give the first polynomial-time constant-factor approximation of the Gromov-Hausdorff distance d_GH between finite point sets in the Euclidean plane; in fixed Euclidean dimension such an approximation was previously known only on the line (Majhi, Vitter and Wenk, 2024). Global alignment cannot succeed: the classical dimension drop defeats alignment by isometries, a multiplicity gap defeats alignment by bijections, and a reflection barrier defeats sorting under any single global reflection pattern. The algorithm is therefore local. Guessing the images of one diameter pair pins every point's longitudinal coordinate to within O(d_GH). Heights are read in windows whose height spread is at most a fixed multiple of their length, where a chain argument makes every compatible match local in the plane. One reflection sign per window is then chosen by 2-SAT; at the right frame and guess, any solution yields a correspondence of distortion O(d_GH). For the bijective relative of d_GH, half the least additive distortion over bijections, the same scheme reduces the planar problem to a matching question that we leave open.

cs.CG↗
arXiv2026

Witness Set in Weak Visibility Polygons is Polynomial-Time Solvable

In the classical Art Gallery Problem (AGP), guards are placed in a polygon so that together they see every point. The Witness Set Problem (WSP), introduced by Amit, Mitchell, and Packer, is a natural dual to the AGP. In this paper, we study the WSP in weak visibility polygons (WVPs), the simple polygons in which every point is seen from some point of one fixed edge. A witness set is a set of points whose visibility regions are pairwise disjoint, so that no single guard sees two of them. A maximum witness set, therefore, lower-bounds the guard number. Exact polynomial-time algorithms for the WSP are known only for monotone mountains, a proper subclass of WVPs. We give the first exact polynomial-time algorithms for the WSP in WVPs, in two settings. In the Discrete Witness Set Problem (DiscWSP), the witnesses come from a given set of $m$ points, and we find a maximum witness subset in $O(n + m \log(n+m))$ time on an $n$-vertex polygon. The algorithm rests on a structural fact: the visibility intersection graph of a WVP, in which two points are adjacent if their visibility regions intersect, is a trapezoid graph, that is, an intersection graph of trapezoids between two parallel lines. Moreover, the class of these graphs properly contains the interval graphs and the permutation graphs, which may be of independent interest in graph theory. We also prove an $Ω(n \log n)$ lower bound in the algebraic decision-tree model for instances with $m = Θ(n)$, so our algorithm for DiscWSP is optimum. In the Continuous Witness Set Problem (ContWSP), a witness may be any point of the polygon, and we give an exact algorithm running in $O(n \log n + ρ^{2}(n + ρ^{2}))$ time, where $ρ$ is the number of reflex vertices.

cs.CG↗
arXiv2026

Bichromatic Line-Centers for Point Pairs

We study the \emph{bichromatic line-center problem} for $n$ pairs of points in the plane. A feasible solution assigns one point from each pair to the red set $R$ and the other to the blue set $B$. The goal is to minimize $\max\{w^\circ(R),\,w^\circ(B)\}$, where $w^\circ(X)$ denotes the minimum width of a strip enclosing $X$; the midlines of the corresponding optimal strips define the line-centers of $R$ and $B$. We consider several variants induced by orientational constraints on line-centers and provide efficient algorithms for each. For one line-center, which consists of computing a minimum-width strip that contains at least one point from each pair, we give an $O(n^2)$-time algorithm. For two line-centers, we obtain an $O(n)$-time algorithm when both are horizontal, and $Θ(n\log n)$-time algorithms when the two centers are parallel or when both orientations are prescribed. When exactly one orientation is prescribed, we give an $O(n^2)$-time algorithm. Finally, for the unrestricted case, we present an $O(n^3\log n)$-time algorithm.

cs.CG↗