Section: New Results

Comparative approximability of hybridization number and directed feedback vertex set

We showed 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. However, despite considerable attention from the combinatorial optimisation community, it remains to this day unknown whether a constant factor polynomial-time approximation exists for DFVS. Our result thus placed the (in)approximability of hybridization number in a much broader complexity context, and as a consequence we obtained that hybridization number inherits inapproximability results from the problem Vertex Cover [16] . On the positive side, we used results from the DFVS literature to give an $O\left(logrloglogr\right)$ approximation for the hybridization number, where $r$ is the value of an optimal solution to the hybridization number problem. This work is submitted for publication.