### Abstract

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. 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 it 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 the hybridization number where r is the correct value.

Original language | English |
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Pages (from-to) | 1635-1656 |

Number of pages | 22 |

Journal | Siam Journal on Discrete Mathematics |

Volume | 26 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2012 |

## Cite this

Kelk, S., van Iersel, L., Lekic, N., Linz, S., Scornavacca, C., & Stougie, L. (2012). Cycle killer... qu'est-ce que c'est? On the comparative approximability of hybridization number and directed feedback vertex set.

*Siam Journal on Discrete Mathematics*,*26*(4), 1635-1656. https://doi.org/10.1137/120864350