Cycle Killer...Qu'est-ce que c'est? On the Comparative Approximability of Hybridization Number and Directed Feedback Vertex Set

@article{Kelk2012CycleKQ,
  title={Cycle Killer...Qu'est-ce que c'est? On the Comparative Approximability of Hybridization Number and Directed Feedback Vertex Set},
  author={Steven M. Kelk and Leo van Iersel and Nela Lekic and Simone Linz and C{\'e}line Scornavacca and Leen Stougie},
  journal={SIAM J. Discret. Math.},
  year={2012},
  volume={26},
  pages={1635-1656}
}
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… 

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References

SHOWING 1-10 OF 48 REFERENCES

On the Complexity of Comparing Evolutionary Trees

Computing the minimum number of hybridization events for a consistent evolutionary history

Computing the Hybridization Number of Two Phylogenetic Trees Is Fixed-Parameter Tractable

TLDR
This paper shows that the problem is fixed-parameter tractable in the two-tree instance when parameterized by this smallest number of reticulation vertices, which is NP-hard even when the collection consists of only two rooted binary phylogenetic trees.

A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem

TLDR
A simple and efficient approximation algorithm with performance ratio of at most 2 is presented, improving previous best bounds for either weighted or unweighted cases of the vertex cover problem.

Quantifying Hybridization in Realistic Time

TLDR
A new fixed-parameter algorithm for computing the minimum number of hybridization events for when two rooted binary phylogenetic trees are given is given, based on interleaving-a technique using repeated kernelization steps that are applied throughout the exhaustive search part of a fixed- parameter algorithm.

Approximating Minimum Feedback Sets and Multicuts in Directed Graphs

TLDR
A combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set, and a generalization of these problems, in which the feedback set has to intersect only a subset of the directed cycles in the graph.

Vertex cover might be hard to approximate to within 2-epsilon

Optimization, approximation, and complexity classes

TLDR
A natural variant of NP, MAX NP, and also a subclass called MAX SNP are defined, which contain several natural, well-studied classes of optimization problems, and it is shown that problems in these classes can be approximated with some bounded error.

Fast FPT Algorithms for Computing Rooted Agreement Forests: Theory and Experiments

TLDR
This work improves on earlier FPT algorithms for computing a rooted maximum agreement forest (MAF) or a maximum acyclic agreement Forest (MAAF) of a pair of phylogenetic trees and introduces new branching rules that reduce the running time of the algorithms.