Constrained and Ordered Level Planarity Parameterized by the Number of Levels

TL;DR

It turns out that CLP is NP-hard even when restricted to instances of height 4, and it is complemented by showing that CLP can be solved in polynomial time for instances of height at most 3.

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.

Figures (20)

• Figure 1: An exhaustive sweeping sequence using all edges of the depicted graph and the corresponding (cf. \ref{['lem:sweep-algo']}) ordered level planar drawing, as well as the corresponding sequence of true dynamic programming table entries $T[s,U]$ (cf. \ref{['lem:algo-exhaustive-sequence']}). Note that in any ordered level planar drawing of the graph, exactly one of the edges $(2,4),(1,3)$ is located to the left of the path $(1,2,3,4)$, while the other is located to the right. Similarly, in any exhaustive sweeping sequence containing separation $s_5$, exactly one of the edges $(2,4),(1,3)$ is used by a separation preceding $s_5$, while the other is used by a separation succeeding $s_5$. Hence, when iteratively building an exhaustive sweeping sequence, it is key to remember which edges between vertices of the current separation have already been used -- this is exactly the purpose of the sets $U$. E.g., the fact that $(2,4)\in U_5$ corresponds to $(2,4)$ being used before $s_5$ (in $s_4$), from which one can infer how to proceed.
• Figure 2: Plugs and sockets; (a) a $(1,3,5,7)$-plug, (b) a $(2,3,5,6)$-socket, (c) a degenerate $(3,3,5,5)$-plug, (d) a degenerate $(2,4,4,6)$-socket, (e) a $(1,3,5,7)$-plug that is linked to a degenerate $(2,4,4,6)$-socket. Connecting vertices are filled in black. Note how in degenerate gadgets ((c) and (d)), the edges and vertices of the repeated levels are contracted.
• Figure 3: Sketches for the proof of \ref{['lem:two_plugs_one_socket']}. The edges of the socket $S$ are bold.
• Figure 4: Example of our parameterized reduction from Multicolored Independent Set (MCIS) to Ordered Level Planarity. On the left side, there is an instance of MCIS with $k=5$ colors. On the right side, there is the schematized grid structure of the ordered level graph constructed from the MCIS instance. The orange disks in the edge blocks represent the places where the collision sockets are placed. Here, the solution $\{v_2, v_3, v_6, v_7, v_8\}$ for the MCIS instances is found.
• Figure 5: Plugs and sockets drawn on color band levels. Sockets are placed in cells. (a) color plug, (b) high plug, (c) pass-through plug, (d) choice socket, (e) color socket, (f) pass-through socket.

Theorems & Definitions (79)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Claim 1
• Claim 2
• Lemma 3
• proof