Parameterized and Approximation Algorithms for the Maximum Bimodal Subgraph Problem

TL;DR

This paper studies the Maximum Bimodal Subgraph problem in plane digraphs, introducing fixed-parameter tractable algorithms and polynomial kernels, leading to efficient approximation schemes and subexponential algorithms.

Contribution

It presents the first fixed-parameter tractable algorithm and polynomial kernel for the MBS problem, advancing the understanding of its parameterized complexity.

Findings

FPT algorithm parameterized by branchwidth/treewidth

Polynomial kernel for the number of non-bimodal vertices

Subexponential FPT algorithm and PTAS for MBS

Abstract

A vertex of a plane digraph is bimodal if all its incoming edges (and hence all its outgoing edges) are consecutive in the cyclic order around it. A plane digraph is bimodal if all its vertices are bimodal. Bimodality is at the heart of many types of graph layouts, such as upward drawings, level-planar drawings, and L-drawings. If the graph is not bimodal, the Maximum Bimodal Subgraph (MBS) problem asks for an embedding-preserving bimodal subgraph with the maximum number of edges. We initiate the study of the MBS problem from the parameterized complexity perspective with two main results: (i) we describe an FPT algorithm parameterized by the branchwidth (and hence by the treewidth) of the graph; (ii) we establish that MBS parameterized by the number of non-bimodal vertices admits a polynomial kernel. As the byproduct of these results, we obtain a subexponential FPT algorithm and an efficient polynomial-time approximation scheme for MBS.