Tree Containment Parameterized by Scanwidth

2026-05-29

AI Summary

This study addresses the TREE CONTAINMENT problem, which asks whether a given rooted phylogenetic tree can be embedded into a rooted phylogenetic network. Introducing the directed graph width measure scanwidth for the first time in this context, the authors design a parameterized algorithm with running time $O(4^{k + k \log k} n + n m^2)$, where $k$ denotes the scanwidth of the network. Furthermore, under the Exponential Time Hypothesis (ETH), they establish a matching lower bound by proving that no algorithm can solve the problem in time $2^{o(c \log c)} n^{O(1)}$, where $c$ is the directed cutwidth. By integrating parameterized complexity, directed width measures, and fine-grained complexity analysis, this work highlights the critical role of the network’s “tree-like” structure in determining computational tractability.

Abstract

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 $k$.

Problem

• Tree Containment
• phylogenetic network
• scanwidth
• NP-complete
• tree-likeness

Innovation

• scanwidth
• tree containment
• parameterized algorithm
• Exponential-Time Hypothesis
• phylogenetic network