Block Crossings in One-Sided Tanglegrams

TL;DR

This paper studies the problem of minimizing block crossings in one-sided tanglegrams, providing complexity results, approximation algorithms, and fixed-parameter algorithms, with initial insights into non-binary trees.

Contribution

It offers a comprehensive analysis of the algorithmic complexity of minimizing block crossings in one-sided tanglegrams, including NP-completeness, approximations, and fixed-parameter algorithms.

Findings

NP-completeness of the problem

Constant-factor approximation algorithms

Fixed-parameter tractable algorithms

Abstract

Tanglegrams are drawings of two rooted binary phylogenetic trees and a matching between their leaf sets. The trees are drawn crossing-free on opposite sides with their leaf sets facing each other on two vertical lines. Instead of minimizing the number of pairwise edge crossings, we consider the problem of minimizing the number of block crossings, that is, two bundles of lines crossing each other locally. With one tree fixed, the leaves of the second tree can be permuted according to its tree structure. We give a complete picture of the algorithmic complexity of minimizing block crossings in one-sided tanglegrams by showing NP-completeness, constant-factor approximations, and a fixed-parameter algorithm. We also state first results for non-binary trees.