The Complexity of Rooted Phylogeny Problems

TL;DR

This paper analyzes the computational complexity of rooted phylogeny problems, identifying tractable cases with efficient algorithms and proving NP-completeness for other restrictions, thus mapping the problem's complexity landscape.

Contribution

It systematically classifies the complexity of rooted phylogeny restrictions, provides efficient algorithms for certain cases, and proves NP-completeness for others, extending understanding of phylogenetic reconstruction problems.

Findings

Polynomial-time algorithm for certain restricted disjunctions of triples

NP-completeness for all other restrictions outside the tractable class

Rooted triple consistency cannot be solved by Datalog

Abstract

Several computational problems in phylogenetic reconstruction can be formulated as restrictions of the following general problem: given a formula in conjunctive normal form where the literals are rooted triples, is there a rooted binary tree that satisfies the formula? If the formulas do not contain disjunctions, the problem becomes the famous rooted triple consistency problem, which can be solved in polynomial time by an algorithm of Aho, Sagiv, Szymanski, and Ullman. If the clauses in the formulas are restricted to disjunctions of negated triples, Ng, Steel, and Wormald showed that the problem remains NP-complete. We systematically study the computational complexity of the problem for all such restrictions of the clauses in the input formula. For certain restricted disjunctions of triples we present an algorithm that has sub-quadratic running time and is asymptotically as fast as the fastest known algorithm for the rooted triple consistency problem. We also show that any restriction of the general rooted phylogeny problem that does not fall into our tractable class is NP-complete, using known results about the complexity of Boolean constraint satisfaction problems. Finally, we present a pebble game argument that shows that the rooted triple consistency problem (and also all generalizations studied in this paper) cannot be solved by Datalog.