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.

翻译：缠绕图描绘了两棵有根二叉系统发育树及其叶集之间的匹配。两棵树分别绘制在对立两侧且无交叉，其叶集在两条垂直线上相对排列。本文不关注最小化成对边交叉数量，而是考虑最小化块交叉（即两条线束在局部相互交叉）的问题。当一棵树固定时，第二棵树的叶子可根据其树结构进行排列。我们通过证明NP完全性、常数因子近似算法以及固定参数算法，完整揭示了最小化单侧缠绕图中块交叉的算法复杂性。此外，我们还首次给出了非二叉树的相关结果。