Constrained Level Planarity is FPT with Respect to the Vertex Cover Number

TL;DR

This paper proves that the Constrained Level Planarity problem is fixed-parameter tractable when parameterized by the vertex cover number of the graph, providing a significant complexity result for this challenging graph drawing problem.

Contribution

The paper establishes that Constrained Level Planarity is solvable in FPT-time with respect to the vertex cover number, resolving an open complexity question.

Findings

Constrained Level Planarity is FPT with respect to vertex cover number.

Previous NP-hardness results hold even for bounded tree-depth and feedback vertex set.

The FPT result is tight; no polynomial-time algorithm is possible under current assumptions.

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, each level y is equipped with a partial order <_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 <_y. Constrained Level Planarity is known to be a remarkably difficult problem: previous results by Klemz and Rote [ACM Trans. Alg. 2019] and by Br\"uckner and Rutter [SODA 2017] imply that it remains NP-hard even when restricted to graphs whose tree-depth and feedback vertex set number are bounded by a constant and even when the instances are additionally required to be either proper, meaning that each edge spans two consecutive levels, or ordered, meaning that all given partial orders are total orders. In particular, these results rule out the existence of FPT-time (even XP-time) algorithms with respect to these and related graph parameters (unless P=NP). However, the parameterized complexity of Constrained Level Planarity with respect to the vertex cover number of the input graph remained open. In this paper, we show that Constrained Level Planarity can be solved in FPT-time when parameterized by the vertex cover number. In view of the previous intractability statements, our result is best-possible in several regards: a speed-up to polynomial time or a generalization to the aforementioned smaller graph parameters is not possible, even if restricting to proper or ordered instances.