88 research outputs found
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
The agreement distance of unrooted phylogenetic networks
A rearrangement operation makes a small graph-theoretical change to a
phylogenetic network to transform it into another one. For unrooted
phylogenetic trees and networks, popular rearrangement operations are tree
bisection and reconnection (TBR) and prune and regraft (PR) (called subtree
prune and regraft (SPR) on trees). Each of these operations induces a metric on
the sets of phylogenetic trees and networks. The TBR-distance between two
unrooted phylogenetic trees and can be characterised by a maximum
agreement forest, that is, a forest with a minimum number of components that
covers both and in a certain way. This characterisation has
facilitated the development of fixed-parameter tractable algorithms and
approximation algorithms. Here, we introduce maximum agreement graphs as a
generalisations of maximum agreement forests for phylogenetic networks. While
the agreement distance -- the metric induced by maximum agreement graphs --
does not characterise the TBR-distance of two networks, we show that it still
provides constant-factor bounds on the TBR-distance. We find similar results
for PR in terms of maximum endpoint agreement graphs.Comment: 23 pages, 13 figures, final journal versio
A first step towards computing all hybridization networks for two rooted binary phylogenetic trees
Recently, considerable effort has been put into developing fast algorithms to
reconstruct a rooted phylogenetic network that explains two rooted phylogenetic
trees and has a minimum number of hybridization vertices. With the standard
approach to tackle this problem being combinatorial, the reconstructed network
is rarely unique. From a biological point of view, it is therefore of
importance to not only compute one network, but all possible networks. In this
paper, we make a first step towards approaching this goal by presenting the
first algorithm---called allMAAFs---that calculates all
maximum-acyclic-agreement forests for two rooted binary phylogenetic trees on
the same set of taxa.Comment: 21 pages, 5 figure
Cycle killer... qu'est-ce que c'est? On the comparative approximability of hybridization number and directed feedback vertex set
We show that the problem of computing the hybridization number of two rooted
binary phylogenetic trees on the same set of taxa X has a constant factor
polynomial-time approximation if and only if the problem of computing a
minimum-size feedback vertex set in a directed graph (DFVS) has a constant
factor polynomial-time approximation. The latter problem, which asks for a
minimum number of vertices to be removed from a directed graph to transform it
into a directed acyclic graph, is one of the problems in Karp's seminal 1972
list of 21 NP-complete problems. However, despite considerable attention from
the combinatorial optimization community it remains to this day unknown whether
a constant factor polynomial-time approximation exists for DFVS. Our result
thus places the (in)approximability of hybridization number in a much broader
complexity context, and as a consequence we obtain that hybridization number
inherits inapproximability results from the problem Vertex Cover. On the
positive side, we use results from the DFVS literature to give an O(log r log
log r) approximation for hybridization number, where r is the value of an
optimal solution to the hybridization number problem
A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest
We give a 2-approximation algorithm for the Maximum Agreement Forest problem
on two rooted binary trees. This NP-hard problem has been studied extensively
in the past two decades, since it can be used to compute the Subtree
Prune-and-Regraft (SPR) distance between two phylogenetic trees. Our result
improves on the very recent 2.5-approximation algorithm due to Shi, Feng, You
and Wang (2015). Our algorithm is the first approximation algorithm for this
problem that uses LP duality in its analysis
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