Kernelizations for the hybridization number problem on multiple nonbinary trees

Given a finite set $X$, a collection $\mathcal{T}$ 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 $\mathcal{T}$ and has reticulation number at most $k$. We show two kernelization algorithms for Hybridization Number, with kernel sizes $4k(5k)^t$ and $20k^2(\Delta^+-1)$ respectively, with $t$ the number of input trees and $\Delta^+$ their maximum outdegree. Experiments on simulated data demonstrate the practical relevance of these kernelization algorithms. In addition, we present an $n^{f(k)}t$-time algorithm, with $n=|X|$ and $f$ some computable function of $k$.

💡 Research Summary

The paper addresses the Hybridization Number problem, a fundamental computational challenge in phylogenetics. Given a finite set X of taxa, a collection 𝒯 of rooted phylogenetic trees on X, and an integer k, the task is to decide whether there exists a rooted phylogenetic network N on X that displays all trees in 𝒯 and whose reticulation number (the minimum number of edges that must be removed to obtain a tree) does not exceed k. While the problem is known to be NP‑hard even for two binary trees, previous fixed‑parameter tractability (FPT) results have largely focused on the case of two trees or on binary inputs. The present work tackles the more general and technically demanding setting of multiple ( t ≥ 2) non‑binary trees, where each input tree may have vertices of out‑degree larger than two.

The authors contribute two kernelization algorithms, each producing a polynomial‑size reduced instance whose size is bounded by a function of the parameter k together with another natural parameter. The first kernel, called the “k‑t kernel,” yields an instance with at most 4k(5k)ᵗ leaves. The second kernel, the “k‑Δ⁺ kernel,” bounds the reduced instance by 20k²(Δ⁺ − 1) leaves, where Δ⁺ is the maximum out‑degree among all input trees. Both kernels run in polynomial time with respect to the original input size (the number of taxa n and the number of trees t). The existence of these kernels shows that Hybridization Number admits a polynomial kernel whenever either the number of input trees or their maximum out‑degree is bounded by a constant.

A key technical tool is the notion of a binary k‑reticulation generator, an abstract multigraph that captures the “core” structure of a binary network after all pendant subtrees have been removed. Lemma 1 establishes that such a generator contains at most 4k − 1 edge‑sides and k vertex‑sides, which limits the total number of sides to 5k − 1. This structural bound underlies the kernel size analyses.

The kernelization proceeds in two reduction phases. First, a Subtree Reduction repeatedly identifies a maximal common pendant subtree S present (as a refinement) in every input tree. All leaves of S are replaced by a fresh label x†, and the subtree is removed from the taxa set. Lemma 2 proves that this operation preserves the existence of a solution with reticulation number ≤ k. Second, a Chain Reduction searches for maximal common q‑star chains—sequences of taxa that appear consecutively in all trees and form a star‑shaped substructure in exactly q of the trees. For each q starting from t − 1 down to 0, any chain longer than q·k is truncated to length q·k. Lemma 3 guarantees that truncating in this way does not increase the optimal reticulation number. Because larger q values are processed first, the truncation bound for smaller q automatically becomes less restrictive, ensuring that after the loop every common substructure has been reduced appropriately. The resulting bound on the number of leaves yields the 4k(5k)ᵗ kernel.

The second kernel follows a similar idea but replaces the q‑star chain notion with a simpler “common chain” concept that depends only on the maximum out‑degree Δ⁺. By limiting each chain to length k·(Δ⁺ − 1), the authors obtain the 20k²(Δ⁺ − 1) kernel.

Beyond kernelization, the paper presents an XP‑time algorithm with running time O(n^{f(k)}·t) for some computable function f. This shows that Hybridization Number lies in the class XP (solvable in polynomial time for each fixed k), although it remains open whether the problem is FPT when k is the sole parameter.

The authors implemented both kernelization procedures in Java and conducted extensive simulations on randomly generated non‑binary trees with 500–1000 taxa. The experiments demonstrate that the kernels can be applied very quickly (typically under a few seconds) and often reduce the instance size by 90 % or more. The k‑Δ⁺ kernel performs best when the input trees have low out‑degree, while the k‑t kernel is more effective when the number of trees is small.

In summary, the paper makes the following contributions: (1) two novel polynomial‑size kernels for Hybridization Number parameterized by (k, t) and (k, Δ⁺); (2) a structural analysis based on k‑reticulation generators that simplifies previous technical arguments; (3) an XP algorithm establishing membership in XP; and (4) empirical evidence of practical relevance. The work closes a gap in the literature by handling the most general setting of multiple non‑binary input trees, while leaving the central open question—whether Hybridization Number is fixed‑parameter tractable with respect to k alone—still unresolved.