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Given a finite set $X$, a collection $\mathcal{T}$ of rooted phylogenetictrees on $X$ and an integer $k$, the Hybridization Number problem asks if thereexists a phylogenetic network on $X$ that displays all trees from $\mathcal{T}$and has reticulation number at most $k$. We show two kernelization algorithmsfor Hybridization Number, with kernel sizes $4k(5k)^t$ and $20k^2(\Delta^+-1)$respectively, with $t$ the number of input trees and $\Delta^+$ their maximumoutdegree. Experiments on simulated data demonstrate the practical relevance ofthese kernelization algorithms. In addition, we present an $n^{f(k)}t$-timealgorithm, with $n=|X|$ and $f$ some computable function of $k$.

Kernelizations for the hybridization number problem on multiple nonbinary trees