Tree Containment Parameterized by Scanwidth
This work provides a new algorithm and complexity bounds for the TREE CONTAINMENT problem, which is important for researchers in mathematical phylogenetics, especially those working with tree-like networks.
This paper addresses the TREE CONTAINMENT problem in mathematical phylogenetics, which determines if a phylogenetic tree can be embedded in a phylogenetic network. The authors developed a parameterized algorithm that solves the problem in O(4^(k + klogk) n + nm^2) time, where k is the scanwidth, and proved a matching lower bound under the Exponential-Time Hypothesis (ETH).
TREE CONTAINMENT is a central decision problem in mathematical phylogenetics, asking whether a given rooted phylogenetic tree is embeddable in ("displayed by") a given rooted phylogenetic network. While the problem is NP-complete for general networks, many algorithmic advances have relied on structural parameters that capture how "tree-like" a network is. In this paper we investigate TREE CONTAINMENT under the structural parameter scanwidth, a directed width measure generalizing popular parameters measuring tree-likeness of phylogenetic networks. We first present a parameterized algorithm that solves the problem in $O(4^{k + k\log{k}} n + nm^2)$ time, where $n$ and $m$ are the numbers of nodes and arcs in the network and $k$ is the width of a given tree-extension. Complementing this upper bound, we prove a matching lower bound under the Exponential-Time Hypothesis (ETH), showing that there is no algorithm for TREE CONTAINMENT that runs in $2^{o(c\log{c})} n^{O(1)}$ time, even on binary inputs, where $c$ is the directed cutwidth of the input network, which upper-bounds the scanwidth $k$.