17 research outputs found
Kernelizations for the hybridization number problem on multiple nonbinary trees
Given a finite set , a collection of rooted phylogenetic
trees on and an integer , the Hybridization Number problem asks if there
exists a phylogenetic network on that displays all trees from
and has reticulation number at most . We show two kernelization algorithms
for Hybridization Number, with kernel sizes and
respectively, with the number of input trees and their maximum
outdegree. Experiments on simulated data demonstrate the practical relevance of
these kernelization algorithms. In addition, we present an -time
algorithm, with and some computable function of
On unrooted and root-uncertain variants of several well-known phylogenetic network problems
The hybridization number problem requires us to embed a set of binary rooted
phylogenetic trees into a binary rooted phylogenetic network such that the
number of nodes with indegree two is minimized. However, from a biological
point of view accurately inferring the root location in a phylogenetic tree is
notoriously difficult and poor root placement can artificially inflate the
hybridization number. To this end we study a number of relaxed variants of this
problem. We start by showing that the fundamental problem of determining
whether an \emph{unrooted} phylogenetic network displays (i.e. embeds) an
\emph{unrooted} phylogenetic tree, is NP-hard. On the positive side we show
that this problem is FPT in reticulation number. In the rooted case the
corresponding FPT result is trivial, but here we require more subtle
argumentation. Next we show that the hybridization number problem for unrooted
networks (when given two unrooted trees) is equivalent to the problem of
computing the Tree Bisection and Reconnect (TBR) distance of the two unrooted
trees. In the third part of the paper we consider the "root uncertain" variant
of hybridization number. Here we are free to choose the root location in each
of a set of unrooted input trees such that the hybridization number of the
resulting rooted trees is minimized. On the negative side we show that this
problem is APX-hard. On the positive side, we show that the problem is FPT in
the hybridization number, via kernelization, for any number of input trees.Comment: 28 pages, 8 Figure
New FPT algorithms for finding the temporal hybridization number for sets of phylogenetic trees
We study the problem of finding a temporal hybridization network for a set of
phylogenetic trees that minimizes the number of reticulations. First, we
introduce an FPT algorithm for this problem on an arbitrary set of binary
trees with leaves each with a running time of , where
is the minimum temporal hybridization number. We also present the concept
of temporal distance, which is a measure for how close a tree-child network is
to being temporal. Then we introduce an algorithm for computing a tree-child
network with temporal distance at most and at most reticulations in
time. Lastly, we introduce a time algorithm for computing a minimum temporal hybridization network for
a set of two nonbinary trees. We also provide an implementation of all
algorithms and an experimental analysis on their performance
A tight kernel for computing the tree bisection and reconnection distance between two phylogenetic trees
In 2001 Allen and Steel showed that, if subtree and chain reduction rules
have been applied to two unrooted phylogenetic trees, the reduced trees will
have at most 28k taxa where k is the TBR (Tree Bisection and Reconnection)
distance between the two trees. Here we reanalyse Allen and Steel's
kernelization algorithm and prove that the reduced instances will in fact have
at most 15k-9 taxa. Moreover we show, by describing a family of instances which
have exactly 15k-9 taxa after reduction, that this new bound is tight. These
instances also have no common clusters, showing that a third
commonly-encountered reduction rule, the cluster reduction, cannot further
reduce the size of the kernel in the worst case. To achieve these results we
introduce and use "unrooted generators" which are analogues of rooted
structures that have appeared earlier in the phylogenetic networks literature.
Using similar argumentation we show that, for the minimum hybridization problem
on two rooted trees, 9k-2 is a tight bound (when subtree and chain reduction
rules have been applied) and 9k-4 is a tight bound (when, additionally, the
cluster reduction has been applied) on the number of taxa, where k is the
hybridization number of the two trees.Comment: One figure added, two small typos fixed. This version to appear in
SIDMA (SIAM Journal on Discrete Mathematics
Cyclic generators and an improved linear kernel for the rooted subtree prune and regraft distance
The rooted subtree prune and regraft (rSPR) distance between two rooted
binary phylogenetic trees is a well-studied measure of topological
dissimilarity that is NP-hard to compute. Here we describe an improved linear
kernel for the problem. In particular, we show that if the classical subtree
and chain reduction rules are augmented with a modified type of chain reduction
rule, the resulting trees have at most 9k-3 leaves, where k is the rSPR
distance; and that this bound is tight. The previous best-known linear kernel
had size O(28k). To achieve this improvement we introduce cyclic generators,
which can be viewed as cyclic analogues of the generators used in the
phylogenetic networks literature. As a corollary to our main result we also
give an improved weighted linear kernel for the minimum hybridization problem
on two rooted binary phylogenetic trees
Embedding Phylogenetic Trees in Networks of Low Treewidth
Given a rooted, binary phylogenetic network and a rooted, binary phylogenetic tree, can the tree be embedded into the network? This problem, called Tree Containment, arises when validating networks constructed by phylogenetic inference methods. We present the first algorithm for (rooted) Tree Containment using the treewidth t of the input network N as parameter, showing that the problem can be solved in 2O(t2) |N| time and space.Optimizatio
New FPT algorithms for finding the temporal hybridization number for sets of phylogenetic trees
We study the problem of finding a temporal hybridization network containing at most k reticulations, for an input consisting of a set of phylogenetic trees. First, we introduce an FPT algorithm for the problem on an arbitrary set of m binary trees with n leaves each with a running time of O(5 k· n· m). We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most d and at most k reticulations in O((8 k) d5 k· k· n· m) time. Lastly, we introduce an O(6 kk! · k· n2) time algorithm for computing a temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance
On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems
International audienceThe hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To thisend we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the âroot uncertainâ variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees