A Tight Subexponential-time Algorithm for Two-Page Book Embedding

TL;DR

This paper presents a nearly optimal subexponential algorithm for two-page graph embeddings, introduces a fixed-parameter algorithm based on treewidth, and proves fixed-parameter tractability with respect to feedback edge number.

Contribution

It provides the first tight subexponential algorithm for two-page book embedding and establishes fixed-parameter tractability for related parameters.

Findings

Achieved a 2^(O(√n)) algorithm for two-page embedding.

Developed a single-exponential FPT algorithm parameterized by treewidth.

Proved fixed-parameter tractability with respect to feedback edge number.

Abstract

A book embedding of a graph is a drawing that maps vertices onto a line and edges to simple pairwise non-crossing curves drawn into pages, which are half-planes bounded by that line. Two-page book embeddings, i.e., book embeddings into 2 pages, are of special importance as they are both NP-hard to compute and have specific applications. We obtain a 2^(O(\sqrt{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.