← 研踪 PaperTrailarXiv:2611.11236 · v1
Algorithms for Soft Tree Containment on Tree-Child Networks
首次提交 2026-11-08 16:46
版本更新 2026-11-08 16:46
雷达观测 2026-11-10 12:16
摘要
Phylogenetic networks generalize evolutionary trees by allowing lineages to merge. The Tree Containment problem asks whether a network $N$ displays a tree $T$ on the same taxa. When the polytomies of $T$ reflect uncertainty rather than simultaneous divergence, the relevant notion is soft display, in which $N$ may refine the polytomies of $T$. We take it in its asymmetric reading. Soft Tree Containment is NP-hard on reticulation-visible networks even at reticulation indegree three, and it was open for a non-binary tree and a tree-child network with multifurcating tree nodes or reticulations of indegree above two. We solve this case. A cluster characterization reduces soft display to realizing all clusters of $T$ by one switching, and in a tree-child network the vertices that can realize a cluster lie on a single chain. The per-cluster requirements then form a constraint network of width one, which arc consistency decides without search, and a $2$-SAT instance when $N$ is binary. This yields, to our knowledge, the first polynomial-time algorithm for soft Tree Containment on single-labelled tree-child networks of arbitrary reticulation indegree, and a linear-time algorithm when $T$ is binary. The same method decides the symmetric variant of the problem, in which the polytomies of $N$ may be resolved as well.
作者备注：38 pages, 8 figures
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v1 · 2026-11-08 16:46 · 最新观测版本
Algorithms for Soft Tree Containment on Tree-Child Networks
Phylogenetic networks generalize evolutionary trees by allowing lineages to merge. The Tree Containment problem asks whether a network $N$ displays a tree $T$ on the same taxa. When the polytomies of $T$ reflect uncertainty rather than simultaneous divergence, the relevant notion is soft display, in which $N$ may refine the polytomies of $T$. We take it in its asymmetric reading. Soft Tree Containment is NP-hard on reticulation-visible networks even at reticulation indegree three, and it was open for a non-binary tree and a tree-child network with multifurcating tree nodes or reticulations of indegree above two. We solve this case. A cluster characterization reduces soft display to realizing all clusters of $T$ by one switching, and in a tree-child network the vertices that can realize a cluster lie on a single chain. The per-cluster requirements then form a constraint network of width one, which arc consistency decides without search, and a $2$-SAT instance when $N$ is binary. This yields, to our knowledge, the first polynomial-time algorithm for soft Tree Containment on single-labelled tree-child networks of arbitrary reticulation indegree, and a linear-time algorithm when $T$ is binary. The same method decides the symmetric variant of the problem, in which the polytomies of $N$ may be resolved as well.
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