301 research outputs found
Phylogenetic incongruence through the lens of Monadic Second Order logic
International audienceWithin the field of phylogenetics there is growing interest in measures for summarising the dissimilarity, or incongruence, of two or more phylogenetic trees. Many of these measures are NP-hard to compute and this has stimulated a considerable volume of research into fixed parameter tractable algorithms. In this article we use Monadic Second Order logic to give alternative, compact proofs of fixed parameter tractability for several well-known incongruence measures. In doing so we wish to demonstrate the considerable potential of MSOL - machinery still largely unknown outside the algorithmic graph theory community - within phylogenetics. A crucial component of this work is the observation that many measures, when bounded, imply the existence of an agreement forest of bounded size, which in turn implies that an auxiliary graph structure, the display graph, has bounded treewidth. It is this bound on treewidth that makes the machinery of MSOL available for proving fixed parameter tractability
Reconstructing pedigrees: some identifiability questions for a recombination-mutation model
Pedigrees are directed acyclic graphs that represent ancestral relationships
between individuals in a population. Based on a schematic recombination
process, we describe two simple Markov models for sequences evolving on
pedigrees - Model R (recombinations without mutations) and Model RM
(recombinations with mutations). For these models, we ask an identifiability
question: is it possible to construct a pedigree from the joint probability
distribution of extant sequences? We present partial identifiability results
for general pedigrees: we show that when the crossover probabilities are
sufficiently small, certain spanning subgraph sequences can be counted from the
joint distribution of extant sequences. We demonstrate how pedigrees that
earlier seemed difficult to distinguish are distinguished by counting their
spanning subgraph sequences.Comment: 40 pages, 9 figure
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
Computing Optimal Leaf Roots of Chordal Cographs in Linear Time
A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with
leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G
if and only if the distance between x and y in T is at most k. Such a tree T is
called a k-leaf root of G. The computational problem of constructing a k-leaf
root for a given graph G and an integer k, if any, is motivated by the
challenge from computational biology to reconstruct phylogenetic trees. For
fixed k, Lafond [SODA 2022] recently solved this problem in polynomial time.
In this paper, we propose to study optimal leaf roots of graphs G, that is,
the k-leaf roots of G with minimum k value. Thus, all k'-leaf roots of G
satisfy k <= k'. In terms of computational biology, seeking optimal leaf roots
is more justified as they yield more probable phylogenetic trees. Lafond's
result does not imply polynomial-time computability of optimal leaf roots,
because, even for optimal k-leaf roots, k may (exponentially) depend on the
size of G. This paper presents a linear-time construction of optimal leaf roots
for chordal cographs (also known as trivially perfect graphs). Additionally, it
highlights the importance of the parity of the parameter k and provides a
deeper insight into the differences between optimal k-leaf roots of even versus
odd k.
Keywords: k-leaf power, k-leaf root, optimal k-leaf root, trivially perfect
leaf power, chordal cographComment: 22 pages, 2 figures, full version of the FCT 2023 pape
Polynomial kernels for 3-leaf power graph modification problems
A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are
V and such that (u,v) is an edge iff u and v are at distance at most 3 in T.
The 3-leaf power graph edge modification problems, i.e. edition (also known as
the closest 3-leaf power), completion and edge-deletion, are FTP when
parameterized by the size of the edge set modification. However polynomial
kernel was known for none of these three problems. For each of them, we provide
cubic kernels that can be computed in linear time for each of these problems.
We thereby answer an open problem first mentioned by Dom, Guo, Huffner and
Niedermeier (2005).Comment: Submitte
Probability, Trees and Algorithms
The subject of this workshop were probabilistic aspects of algorithms for fundamental problems such as sorting, searching, selecting of and within data, random permutations, algorithms based on combinatorial trees or search trees, continuous limits of random trees and random graphs as well as random geometric graphs. The deeper understanding of the complexity of such algorithms and of shape characteristics of large discrete structures require probabilistic models and an asymptotic analysis of random discrete structures. The talks of this workshop focused on probabilistic, combinatorial and analytic techniques to study asymptotic properties of large random combinatorial structures
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