[Paper Review] Compatibility of Partitions, Hierarchies, and Split Systems.

Marc Hellmuth, David Schaller+1 more|arXiv (Cornell University)|Apr 29, 2021

Rough Sets and Fuzzy Logic14 references4 citations

TL;DR

This paper presents characterizations and a linear-time algorithm to determine compatibility between a partition and a hierarchy or tree-like split system, particularly focusing on whether a refined tree exists that makes the partition compatible. It shows that compatibility with a single partition is solvable in linear time, but becomes NP-complete when multiple partitions are involved—though fixed-parameter tractable when parameterized by the number of 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. We report several characterizations for (refinements of) hierarchies and split systems that are compatible with (sets 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 set of partitions is considered instead of a single partition. We finally explore the close relationship of the concept of compatibility and so-called Fitch maps.

Motivation & Objective

• To determine when a partition is compatible with a hierarchy or split system in phylogenetic classification.
• To characterize conditions under which a refinement of a tree exists such that the partition aligns with leaf sets of connected components.
• To explore the computational complexity of compatibility when multiple partitions are considered.
• To establish a connection between compatibility and Fitch maps, a known concept in phylogenetic reconstruction.

Proposed method

• Proposes a characterization of compatibility between a partition and a hierarchy or split system using structural properties of trees and edge deletions.
• Introduces a linear-time algorithm to check compatibility between a single partition and a refined tree representation of a hierarchy or split system.
• Employs tree decomposition and edge-removal analysis to identify connected components whose leaf sets match partition elements.
• Applies fixed-parameter tractability techniques to handle the NP-complete case when multiple partitions are considered.
• Leverages the theory of Fitch maps to establish structural and algorithmic connections between compatibility and map-based constraints.

Experimental results

Research questions

• RQ1 Under what conditions is a partition compatible with a given hierarchy or split system?
• RQ2 Can a refinement of a tree be constructed such that the partition's sets correspond to the leaf sets of connected components after edge deletions?
• RQ3 What is the computational complexity of checking compatibility when multiple partitions are involved?
• RQ4 How are compatibility relations related to Fitch maps in the context of phylogenetic trees?

Key findings

• A linear-time algorithm exists to check compatibility between a single partition and a hierarchy or split system.
• Compatibility with a single partition is solvable in linear time, indicating efficient tractability for practical applications.
• When multiple partitions are considered, the problem becomes NP-complete, highlighting increased computational difficulty.
• Despite NP-completeness, the problem remains fixed-parameter tractable when parameterized by the number of partitions.
• A close structural relationship is established between compatibility and Fitch maps, enabling new algorithmic insights.