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.
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