The Complexity of Phylogeny Constraint Satisfaction Problems

TL;DR

This paper classifies the computational complexity of a broad class of phylogenetic reconstruction problems, showing they are either solvable in polynomial time or NP-complete, and provides a polynomial-time algorithm for the tractable cases.

Contribution

It establishes a complexity dichotomy for phylogeny constraint satisfaction problems and generalizes a polynomial-time algorithm for rooted triple consistency.

Findings

Problems are either polynomial-time solvable or NP-complete.

A generalized polynomial-time algorithm for rooted triple consistency.

A complexity classification applicable to many phylogenetic problems.

Abstract

We systematically study the computational complexity of a broad class of computational problems in phylogenetic reconstruction. The class contains for example the rooted triple consistency problem, forbidden subtree problems, the quartet consistency problem, and many other problems studied in the bioinformatics literature. The studied problems can be described as \emph{constraint satisfaction problems} where the constraints have a first-order definition over the rooted triple relation. We show that every such phylogeny problem can be solved in polynomial time or is NP-complete. On the algorithmic side, we generalize a well-known polynomial-time algorithm of Aho, Sagiv, Szymanski, and Ullman for the rooted triple consistency problem. Our algorithm repeatedly solves linear equation systems to construct a solution in polynomial time. We then show that every phylogeny problem that cannot be solved by our algorithm is NP-complete. Our classification establishes a dichotomy for a large class of infinite structures that we believe is of independent interest in universal algebra, model theory, and topology. The proof of our main result combines results and techniques from various research areas: a recent classification of the model-complete cores of the reducts of the homogeneous binary branching C-relation, Leeb's Ramsey theorem for rooted trees, and universal algebra.