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

S. Kelk, L.J.J. van Iersel, N. Lekic, S. Linz, C. Scornavacca, L. Stougie

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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. Copyright © by SIAM.
Original languageEnglish
Pages (from-to)1635-1656
JournalSIAM Journal on Discrete Mathematics
Volume26
Issue number4
DOIs
Publication statusPublished - 2012

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