Compatibility of Partitions with Trees, Hierarchies, and Split Systems

TL;DR

This paper investigates the compatibility between partitions, hierarchies, and trees in classification problems, providing characterizations and efficient algorithms, including a linear-time method for certain cases and fixed-parameter tractability results.

Contribution

It offers new characterizations of compatibility and introduces a linear-time algorithm for checking compatibility with trees and partitions, along with complexity results for multiple partitions.

Findings

Provided characterizations for compatible hierarchies and split systems.

Developed a linear-time algorithm for compatibility checking.

Established fixed-parameter tractability for multiple partitions.

Abstract

The question whether a partition $\mathcal{P}$ and a hierarchy $\mathcal{H}$ or a tree-like split system $\mathfrak{S}$ are compatible naturally arises in a wide range of classification problems. In the setting of phylogenetic trees, one asks whether the sets of $\mathcal{P}$coincide with leaf sets of connected components obtained by deleting some edges from the tree $T$ that represents $\mathcal{H}$ or $\mathfrak{S}$, respectively. More generally, we ask whether a refinement $T^*$ of $T$ exists such that $T^*$ and $\mathcal{P}$ are compatible in this sense. The latter is closely related to the question as to whether there exists a tree at all that is compatible with $\mathcal{P}$. We report several characterizations for (refinements of) hierarchies and split systems that are compatible with (systems of) partitions. In addition, we provide a linear-time algorithm to check whether refinements of trees and a given partition are compatible. The latter problem becomes NP-complete but fixed-parameter tractable if a system of partitions is considered instead of a single partition. In this context, we also explore the close relationship of the concept of compatibility and so-called Fitch maps.