It was 14 years ago that the Millennium started, but there are therefore still 986 years left to solve the following seven phylogenetic network Millennium problems. These are not necessarily the most important problems to solve from a biological point of view, but are challenging computational problems that have (at least) some biological relevance. The problems are all about phylogenetic networks, except for Problems 2 and 7 which are about the closely related topic of

*agreement forests*. Solving these problems will

**no**t be rewarded with $1,000,000 but only with eternal fame.

In each of these problems, a

*phylogenetic network*on

*X*is a directed acyclic graph with a single root and no vertices that have only one incoming and only one outgoing arc, and in which each leaf is labelled by an element of

*X*and each element of

*X*labels one leaf

*.*

**Problem 1.**Is the Hybridization Number problem fixed-parameter tractable (FPT) if the input is an unrestricted set of nonbinary trees and the only parameter is the hybridization number? Hybridization Number is the following problem. Given a finite set

*X*, a collection

*T*of rooted (possibly nonbinary) phylogenetic trees on

*X*and a natural number

*k*, decide if there exists a rooted phylogenetic network on

*X*that displays all trees from

*T*and has reticulation number at most

*k*. See e.g. (van Iersel, Kelk, 2013) for more detailed definitions.

**Problem 2.**Does there exist a polynomial-time 2-approximation algorithm for MAF on two binary trees? Maximum Agreement Forest (MAF) on two binary trees can be defined as follows. Given a finite set

*X*and two rooted binary phylogenetic trees on

*X*, what is the minimum number number of components in a forest on

*X*that can be obtained from each of the input trees by deleting vertices, deleting edges and suppressing indegree-1 outdegree-1 vertices? For a 2.5-approximation see (Shi, You, Feng, 2014).

**Problem 3.**Is there an FPT algorithm for finding a level-

*k*phylogenetic network consistent with a given dense set of rooted triplets, if

*k*is the parameter? A

*rooted triplet*is a phylogenetic tree with three leaves. A set of rooted triplets is called

*dense*if it contains at least one triplet for each combination of three leaves. A network is

*level*-

*k*if it can be turned into a tree by deleting at most

*k*edges per biconnected component. This problem is known to be solvable in polynomial time if k is fixed, see (Habib and To 2012).

**Problem 4**. Is Tree Containment polynomial-time solvable or NP-hard for reticulation visible networks? Tree Containment is the problem of deciding if a given phylogenetic network displays a given tree. A phylogenetic network is called

*reticulation visible*if for each reticulation there exists some leaf such that each path from the root to that leaf passes through that reticulation. Tree Containment is known to be NP-hard for general networks and for some restricted classes of networks; see (Kanj, Nakhleh, Than, Xia, 2008) and (van Iersel, Semple, Steel 2010).

**Problem 5.**Is there a constant-factor approximation algorithm for computing the softwired parsimony score of a binary tree-child network and a binary character? Given a network and a character state (0 or 1) for each leaf, the

*softwired parsimony score*is the minimum number of state-changes in any tree (on all leaves) displayed by the network, over all possible assignments of states to the internal vertices. A phylogenetic network is called

*tree-child*if each non-leaf vertex has at least one child that is not a reticulation. This problem does not have a constant factor approximation for general networks or for other (less severely) restricted classes of networks, unless P = NP (Fischer, van Iersel, Kelk, Scornavacca 2013).

**Problem 6.**Given

*k*> 1, what is the maximum value of

*p*such that for any set of rooted triplets there exists some level-

*k*phylogenetic network on

*n*leaves that is consistent with at least a fraction

*p*of the input triplets? For

*k*= 0 the maximum is

*p*= 1/3 and for

*k*= 1 it is roughly 0.48, see (Byrka, Gawrychowski, Huber, Kelk 2009).

**Problem 7**. Is there an

*O(c^n)*algorithm for Maximum Acyclic Agreement Forest (MAAF) on two binary phylogenetic trees with

*c*< 2? An

*acyclic agreement forest*is an agreement forest (see above) for which the following directed graph

*D*is acyclic.

*D*has a vertex for each component of the forest and there is an arc from component

*A*to component

*B*if in at least one of the input trees there is a directed path from the root of

*A*to the root of

*B*. It is known that there exist an

*O*(2^n)*algorithm for this problem (van Iersel, Kelk, Lekic, Stougie, 2013).