The complexity of deriving multi-labeled trees from bipartitions.

Area of Science:

• Phylogenetics and evolutionary biology
• Computational biology and bioinformatics

Background:

• Multi-labeled trees are extensions of phylogenetic trees, accommodating multiple instances of a species.
• Phylogenetic trees are efficiently encoded using bipartitions of their leaf sets.
• Understanding polyploid species' evolutionary history requires advanced tree representations.

Purpose of the Study:

• To investigate the computational complexity of representing multi-labeled trees.
• To generalize existing phylogenetic tree characterization methods for multi-labeled trees.
• To develop an efficient algorithm for analyzing multi-labeled tree representations.

Main Methods:

• Demonstrated the NP-hardness of deciding if a multiset's bipartition collection can form a multi-labeled tree.
• Generalized a known condition for phylogenetic tree encoding to multi-labeled trees.
• Developed a fixed-parameter tractable algorithm for the decision problem.

Main Results:

• The problem of representing multi-labeled trees via bipartitions is computationally difficult (NP-hard).
• A generalized condition for phylogenetic tree encoding was successfully adapted for multi-labeled trees.
• A fixed-parameter algorithm was derived, offering computational efficiency relative to a specific parameter.

Conclusions:

• While representing multi-labeled trees is NP-hard, a generalized characterization and an efficient algorithm exist.
• This work provides a computational framework for studying the evolutionary origins of polyploid species using multi-labeled trees.
• The developed algorithm offers a practical approach for analyzing complex evolutionary histories represented by multisets.