Constrained Level Planarity is Fixed-Parameter Tractable with Respect to the Vertex Cover Number

Core Concepts

Constrained Level Planarity, a generalization of the classical Level Planarity problem, can be solved in fixed-parameter tractable time when parameterized by the vertex cover number of the input graph.

Abstract

The paper studies the Constrained Level Planarity (CLP) problem, which asks whether a given constrained level graph admits a crossing-free drawing where the left-to-right order of vertices on each level respects the given partial orders. The key insights are: CLP is known to be NP-hard even when restricted to graphs with bounded tree-width, path-width, tree-depth, or feedback vertex set number. However, the parameterized complexity of CLP with respect to the vertex cover number remained open. The authors show that CLP can be solved in FPT-time when parameterized by the vertex cover number. This is achieved by: Defining the notion of a "core-induced subdrawing" of a refined visibility extension of a constrained level planar drawing, which captures crucial structural properties while having size bounded by the vertex cover number. Describing an algorithm that first guesses this core-induced subdrawing, then inserts the transition vertices, and finally places the leaves and ears. The algorithm runs in 2^(O(k log k)) * n^O(1) time, where k is the vertex cover number and n is the number of vertices. This is best-possible, as CLP remains NP-hard even for graphs with bounded parameters smaller than the vertex cover number.

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Key Insights Distilled From

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

by Boris Klemz,... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16723.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to solve other variants or generalizations of the Level Planarity problem

The techniques developed in this paper can be extended to solve other variants or generalizations of the Level Planarity problem by adapting the algorithm to accommodate different constraints or additional parameters. For example, one could consider variations where the vertices have different levels of importance or where certain edges have specific requirements for their placement. By modifying the traversal sequences, channel definitions, and insertion sequences to account for these new conditions, the algorithm can be tailored to address a wider range of Level Planarity problems. Additionally, the concept of visibility extensions and cores can be applied to other graph drawing problems that involve constraints on the placement of vertices and edges.

Can the FPT algorithm be further improved in terms of the dependence on the vertex cover number

While the FPT algorithm presented in the paper already shows significant improvement in terms of the dependence on the vertex cover number, further enhancements can be explored. One approach could involve optimizing the enumeration process in Step 1 to reduce the number of possible options for the set of crucial ears and their levels. By refining the search space and implementing more efficient enumeration techniques, the algorithm's runtime complexity could potentially be reduced even further. Additionally, fine-tuning the insertion and placement steps in Steps 2 and 3 could lead to faster execution and improved overall performance.

What other graph parameters, besides the vertex cover number, could lead to efficient algorithms for Constrained Level Planarity

Apart from the vertex cover number, other graph parameters that could lead to efficient algorithms for Constrained Level Planarity include the treewidth, pathwidth, and feedback vertex set number of the input graph. By investigating how these parameters interact with the constraints imposed by the problem, it may be possible to develop specialized algorithms that exploit specific structural properties of the graph to achieve faster computation. Additionally, considering parameters related to the connectivity or density of the graph could offer insights into optimizing the algorithm for different types of input graphs.