Kernelizations for the hybridization number problem on multiple nonbinary trees

Abstract

Given a finite set X, a collection T of rooted phylogenetic trees on X and an integer k, the Hybridization Number problem asks if there exists a phylogenetic network on X that displays all trees from T and has reticulation number at most k. We show two kernelization algorithms for Hybridization Number, with kernel sizes 4k(5k) t and 20k 2 ( + − 1) respectively, with t the number of input trees and + their maximum outdegree. Experiments on simulated data demonstrate the practical relevance of our kernelization algorithms. In addition, we present an n f (k) t-time algorithm, with n = |X| and f some computable function of k.

Parameterized Algorithms in Bioinformatics: An Overview

“…Concerning preprocessing, a kernel with at most 9k taxa is known [188,201] and this kernelization result has been generalized to the case of deciding HN for t > 2 binary trees (in which case HN and MAAF no longer coincide) by van Iersel and Linz [207], showing a kernel with 20k 2 taxa for this case, which has again been generalized to t > 2 non-binary trees by van Iersel et al [208], showing a kernel with at most 20k 2 (∆ − 1) (and at most 4k(5k) t ) taxa [208]. For MAAF with t = 2 non-binary trees, Linz and Semple [203] showed a linear bikernel (that is, a kernelization into a different problem, see [209]) with 89k taxa, which implies a quadratic-size classical kernel.…”

On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems

“…Hence, the overall running time is the time for the kernelization procedure plus calls to an algorithm for HN , where . Noting that (otherwise the answer is trivially NO), and that HN is FPT [ 31 ], we obtain an overall running time of .…”

“…[ 14 , 30 , 37 ]). In [ 5 ] it was proven that HN is FPT (in the hybridization number) for two input trees and in recent years the result has been generalized in a number of directions (see [ 31 ] and the references therein for a recent overview).…”

Cyclic generators and an improved linear kernel for the rooted subtree prune and regraft distance

“…graph); it is a cyclic variant of the much-studied minimum hybridization problem (see e.g. [17] and links therein). This allows us to introduce cyclic generators which summarize the backbone of such networks, and allow us to carefully bound the size of reduced instances.…”