Core Concepts

The article presents a 2O(√n) algorithm for computing a two-page book embedding of an n-vertex graph, which is asymptotically tight under the Exponential Time Hypothesis. The algorithm also includes a single-exponential fixed-parameter algorithm for the same problem when parameterized by the treewidth of the input graph.

Abstract

The article focuses on the problem of computing two-page book embeddings of graphs, which are drawings that map vertices onto a line and edges to simple pairwise non-crossing curves drawn into two "pages".
The key highlights and insights are:
The authors obtain a 2O(√n) algorithm for computing a two-page book embedding of an n-vertex graph, which is asymptotically tight under the Exponential Time Hypothesis.
As a key tool, the authors obtain a single-exponential fixed-parameter algorithm for the same problem when parameterized by the treewidth of the input graph. This improves upon previous results that relied on Courcelle's Theorem.
The authors establish the fixed-parameter tractability of computing minimum-page book embeddings when parameterized by the feedback edge number, settling an open question.
The authors introduce a novel framework for dynamic programming on sphere-cut decompositions that allows them to transfer records from child to parent nooses via XOR operations, which may be of broader interest.

Stats

Every n-vertex graph is known to admit an ⌈n/2⌉-page book embedding.
The class of graphs that can be embedded on two pages is a superclass of planar graphs with maximum degree at most 4.
Testing subhamiltonicity, which is equivalent to determining if a graph admits a two-page book embedding, is an NP-hard problem.

Quotes

"We obtain a 2O(√n) algorithm for computing a book embedding of an n-vertex graph on two pages—a result which is asymptotically tight under the Exponential Time Hypothesis."
"As a key tool in our approach, we obtain a single-exponential fixed-parameter algorithm for the same problem when parameterized by the treewidth of the input graph."
"We conclude by establishing the fixed-parameter tractability of computing minimum-page book embeddings when parameterized by the feedback edge number, settling an open question arising from previous work on the problem."

Key Insights Distilled From

by Robert Gania... at **arxiv.org** 04-23-2024

Deeper Inquiries

The techniques developed in this paper for solving two-page book embeddings can be extended to tackle other graph embedding problems by adapting the algorithm to handle different constraints and requirements specific to the new problem. For instance, the dynamic programming framework on sphere-cut decompositions can be applied to problems where the goal is to embed a graph on a surface with specific properties or constraints. By modifying the types defined for nodes in the SPQR-tree and adjusting the dynamic programming approach, the algorithm can be tailored to address various graph embedding problems beyond two-page book embeddings. Additionally, the concept of normal forms and witness structures can be utilized in different embedding scenarios to ensure the embedding satisfies certain criteria or conditions.

The Exponential Time Hypothesis (ETH) posits that there are problems that cannot be solved significantly faster than the brute-force exponential time algorithms. While the presented algorithm in the paper is asymptotically tight under ETH, future advancements in complexity theory could potentially impact the tightness of the algorithm. If new breakthroughs or techniques are discovered that challenge the assumptions of ETH or provide alternative approaches to solving hard problems, it may lead to improvements in the efficiency and tightness of algorithms for graph embedding and other NP-hard problems. However, until such advancements occur, the algorithm's tightness under ETH provides a strong foundation for its performance in solving two-page book embedding problems.

The authors' framework for dynamic programming on sphere-cut decompositions has several potential applications and implications in the field of parameterized algorithms and graph theory. One key implication is the ability to efficiently solve graph embedding problems parameterized by various graph parameters such as treewidth, feedback edge number, or treedepth. By leveraging the dynamic programming approach on sphere-cut decompositions, researchers can develop algorithms for a wide range of graph-related problems that exhibit structural characteristics suitable for this framework. Additionally, the techniques introduced in the paper could be extended to address other combinatorial optimization problems that can benefit from a similar dynamic programming strategy, leading to advancements in parameterized algorithm design and analysis.

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