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

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. However, 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 hybridization number 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 hybridization number, where r is the value of an optimal solution to the hybridization number problem.

A Tight Kernel for Computing the Tree Bisection and Reconnection Distance between Two Phylogenetic Trees

“…As a spin-off to these results we show that the earlier-identified upper bound of 9k − 2 [21] on the size of the standard hybridization number weighted kernel [7] for rooted trees is also tight, and that in this case the cluster reduction can only improve the bound slightly, to 9k − 4, which as we demonstrate is also tight.…”

“…Generators. Rooted generators have played an important role in establishing kernelization results for problems on rooted trees [21,28,30], but have only received very little attention [14] as a tool to tackle problems on unrooted trees. Here we give a definition of an unrooted generator that we will subsequently use to establish an improved kernel for the problem of computing the TBR distance (formally defined below) between two trees.…”

A First Step Toward Computing All Hybridization Networks For Two Rooted Binary Phylogenetic Trees

“…Now, noting that the subtree and chain reduction can be computed in O(n 3 ) for two rooted binary phylogenetic X -trees, where n = jX j (Bordewich and Semple, 2007a), the next corollary is an immediate consequence of Theorem 5, as well as the kernelization ideas that are presented in Bordewich and Semple (2007a) and Kelk et al (2012) and that are briefly summarized prior to this paragraph.…”

“…More precisely, it is shown in Bordewich and Semple (2007a;Lemma 3.3) that, by repeatedly applying the subtree and chain reductions to S and T until no further reduction is possible, the leaf set size of the soobtained rooted binary phylogenetic trees is at most 14h(S,T). This bound has recently been improved to 9h(S,T) by Kelk et al (2012). It is now straightforward to see that modifying allMAAFs(S,T,R, F , k,M) in the following way is sufficient to make use of this result.…”

Kernelizations for the hybridization number problem on multiple nonbinary trees

“…Even in the case when T consists of two binary (that is, bifurcating) trees the problem is NP-hard, APX-hard [5] and in terms of approximability is a surprisingly ✩ A preliminary version of this article appeared in the Proceedings of Workshop on Graph-Theoretic Concepts in Computer Science (WG 2014). close relative of the problem Directed Feedback Vertex Set [17,13]. On the positive side, this variant of the problem is fixed-parameter tractable (FPT) in parameter k, the reticulation number of an optimal network.…”