826 research outputs found
Hitting and Harvesting Pumpkins
The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges.
A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of
G, each inducing a connected subgraph of G, such that there are at least c
edges in G between A and B. We focus on covering and packing c-pumpkin-models
in a given graph: On the one hand, we provide an FPT algorithm running in time
2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be
covered by at most k vertices. This generalizes known single-exponential FPT
algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the
cases c=1,2 respectively. On the other hand, we present a O(log
n)-approximation algorithm for both the problems of covering all
c-pumpkin-models with a smallest number of vertices, and packing a maximum
number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change
Deep kernelization for the Tree Bisection and Reconnnect (TBR) distance in phylogenetics
We describe a kernel of size 9k-8 for the NP-hard problem of computing the
Tree Bisection and Reconnect (TBR) distance k between two unrooted binary
phylogenetic trees. We achieve this by extending the existing portfolio of
reduction rules with three novel new reduction rules. Two of the rules are
based on the idea of topologically transforming the trees in a
distance-preserving way in order to guarantee execution of earlier reduction
rules. The third rule extends the local neighbourhood approach introduced in
(Kelk and Linz, Annals of Combinatorics 24(3), 2020) to more global structures,
allowing new situations to be identified when deletion of a leaf definitely
reduces the TBR distance by one. The bound on the kernel size is tight up to an
additive term. Our results also apply to the equivalent problem of computing a
Maximum Agreement Forest (MAF) between two unrooted binary phylogenetic trees.
We anticipate that our results will be more widely applicable for computing
agreement-forest based dissimilarity measures.Comment: 38 pages. In this version a figure has been added, some references
have been added, some small typo's have been fixed and the introduction and
conclusion have been slightly extended. Submitted for journal revie
Kernelization of Whitney Switches
A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs G
and H are 2-isomorphic, or equivalently, their cycle matroids are isomorphic,
if and only if G can be transformed into H by a series of operations called
Whitney switches. In this paper we consider the quantitative question arising
from Whitney's theorem: Given two 2-isomorphic graphs, can we transform one
into another by applying at most k Whitney switches? This problem is already
NP-complete for cycles, and we investigate its parameterized complexity. We
show that the problem admits a kernel of size O(k), and thus, is
fixed-parameter tractable when parameterized by k.Comment: To appear at ESA 202
Beyond Adjacency Maximization: Scaffold Filling for New String Distances
International audienceIn Genomic Scaffold Filling, one aims at polishing in silico a draft genome, called scaffold. The scaffold is given in the form of an ordered set of gene sequences, called contigs. This is done by confronting the scaffold to an already complete reference genome from a close species. More precisely, given a scaffold S, a reference genome G and a score function f () between two genomes, the aim is to complete S by adding the missing genes from G so that the obtained complete genome S * optimizes f (S * , G). In this paper, we extend a model of Jiang et al. [CPM 2016] (i) by allowing the insertions of strings instead of single characters (i.e., some groups of genes may be forced to be inserted together) and (ii) by considering two alternative score functions: the first generalizes the notion of common adjacencies by maximizing the number of common k-mers between S * and G (k-Mer Scaffold Filling), the second aims at minimizing the number of breakpoints between S * and G (Min-Breakpoint Scaffold Filling). We study these problems from the parameterized complexity point of view, providing fixed-parameter (FPT) algorithms for both problems. In particular, we show that k-Mer Scaffold Filling is FPT wrt. parameter , the number of additional k-mers realized by the completion of S—this answers an open question of Jiang et al. [CPM 2016]. We also show that Min-Breakpoint Scaffold Filling is FPT wrt. a parameter combining the number of missing genes, the number of gene repetitions and the target distance
Kernelization of Whitney Switches
A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs and are 2-isomorphic, or equivalently, their cycle matroids are isomorphic if and only if can be transformed into by a series of operations called Whitney switches. In this paper we consider the quantitative question arising from Whitney's theorem: Given two 2-isomorphic graphs, can we transform one into another by applying at most Whitney switches? This problem is already \sf NP-complete for cycles, and we investigate its parameterized complexity. We show that the problem admits a kernel of size and thus is fixed-parameter tractable when parameterized by .publishedVersio
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
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