  Phylogenetic networks are often constructed by merging multiple conflicting
phylogenetic signals into a directed acyclic graph. It is interesting to
explore whether a network constructed in this way induces biologically-relevant
phylogenetic signals that were not present in the input. Here we show that,
given a multiple alignment A for a set of taxa X and a rooted phylogenetic
network N whose leaves are labelled by X, it is NP-hard to locate the most
parsimonious phylogenetic tree displayed by N (with respect to A) even when the
level of N - the maximum number of reticulation nodes within a biconnected
component - is 1 and A contains only 2 distinct states. (If, additionally, gaps
are allowed the problem becomes APX-hard.) We also show that under the same
conditions, and assuming a simple binary symmetric model of character
evolution, finding the most likely tree displayed by the network is NP-hard.
These negative results contrast with earlier work on parsimony in which it is
shown that if A consists of a single column the problem is fixed parameter
tractable in the level. We conclude with a discussion of why, despite the
NP-hardness, both the parsimony and likelihood problem can likely be
well-solved in practice.
