  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.
