Steven Kelk,
Leo van Iersel,
Nela Lekic,
Simone Linz,
Celine Scornavacca and
Leen Stougie. Cycle killer... qu'estce que c'est? On the comparative approximability of hybridization number and directed feedback vertex set. In SIDMA, Vol. 26(4):16351656, 2012. Keywords: agreement forest, approximation, explicit network, from rooted trees, minimum number, phylogenetic network, phylogeny, Program CycleKiller, reconstruction. Note: http://arxiv.org/abs/1112.5359, about the title.
Toggle 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 polynomialtime approximation if and only if the problem of computing a minimumsize feedback vertex set in a directed graph (DFVS) has a constant factor polynomialtime 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 NPcomplete problems. Despite considerable attention from the combinatorial optimization community, it remains to this day unknown whether a constant factor polynomialtime 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."
