[Paper Review] Treewidth of display graphs: bounds, brambles and applications

Remie Janssen, Mark Jones+3 more|arXiv (Cornell University)|Sep 4, 2018

Plant and animal studies36 references3 citations

TL;DR

This paper investigates the treewidth of display graphs formed by unrooted binary phylogenetic trees and networks, proving that recognizing such graphs is NP-hard in general but solvable in linear time when treewidth is bounded. It establishes a tight upper bound of $2\text{tw}(N)+1$ on the treewidth of a display graph when a network $N$ displays a tree $T$, and uses this to develop an FPT algorithm parameterized by treewidth for the network-display problem.

ABSTRACT

Phylogenetic trees and networks are leaf-labelled graphs used to model evolution. Display graphs are created by identifying common leaf labels in two or more phylogenetic trees or networks. The treewidth of such graphs is bounded as a function of many common dissimilarity measures between phylogenetic trees and this has been leveraged in fixed parameter tractability results. Here we further elucidate the properties of display graphs and their interaction with treewidth. We show that it is NP-hard to recognize display graphs, but that display graphs of bounded treewidth can be recognized in linear time. Next we show that if a phylogenetic network displays (i.e. topologically embeds) a phylogenetic tree, the treewidth of their display graph is bounded by a function of the treewidth of the original network (and also by various other parameters). In fact, using a bramble argument we show that this treewidth bound is sharp up to an additive term of 1. We leverage this bound to give an FPT algorithm, parameterized by treewidth, for determining whether a network displays a tree, which is an intensively-studied problem in the field. We conclude with a discussion on the future use of display graphs and treewidth in phylogenetics.

Motivation & Objective

• To investigate the structural and algorithmic properties of display graphs formed by phylogenetic trees and networks.
• To address the open problem of recognizing display graphs, which was posed in prior work.
• To establish tight treewidth bounds for display graphs when a phylogenetic network displays a tree.
• To develop fixed-parameter tractable (FPT) algorithms for the network-display problem using treewidth as a parameter.
• To explore the broader applicability of display graphs and treewidth in phylogenetic inference and algorithm design.

Proposed method

• Proves NP-hardness of the display graph recognition problem by reducing it to the TreeArboricity problem.
• Uses Courcelle’s Theorem to show that display graph recognition is linear-time solvable on graphs of bounded treewidth, by expressing the property in Monadic Second-Order Logic (MSOL).
• Introduces a bramble-based argument to prove that the treewidth bound $2\text{tw}(N)+1$ for the display graph of a network $N$ and a tree $T$ that it displays is tight up to an additive constant.
• Develops an MSOL formula to express the condition that a network $N$ displays a tree $T$ via edge deletion, using predicates for connectivity, acyclicity, and quartet compatibility.
• Leverages the treewidth bound to design an FPT algorithm for the network-display problem, parameterized by treewidth.
• Provides alternative treewidth bounds in terms of network level and reticulation number, making the results accessible to phylogeneticists.

Experimental results

Research questions

• RQ1 Is the problem of recognizing whether a given cubic graph is a suppressed display graph NP-hard?
• RQ2 Can display graphs of bounded treewidth be recognized in linear time?
• RQ3 What is the tightest possible upper bound on the treewidth of a display graph formed by a phylogenetic network $N$ and a tree $T$ that $N$ displays?
• RQ4 Can the treewidth bound on the display graph be used to design an FPT algorithm for determining whether a network displays a given tree?
• RQ5 How do parameters like network level and reticulation number relate to the treewidth of the resulting display graph?

Key findings

• The problem of recognizing whether a given cubic graph is a suppressed display graph is NP-hard.
• When the treewidth is bounded, display graph recognition can be solved in linear time using Courcelle’s Theorem and an MSOL formula.
• If a phylogenetic network $N$ displays a tree $T$, the treewidth of their display graph is at most $2\text{tw}(N)+1$, and this bound is tight up to an additive constant of 1.
• Alternative treewidth bounds are provided in terms of the network’s level and reticulation number, which are more familiar to phylogeneticists.
• An FPT algorithm for the network-display problem is developed, parameterized by treewidth, leveraging the established treewidth bound.
• The MSOL formula for tree-display in networks uses predicates for connectivity, acyclicity, and quartet compatibility, and has fixed length independent of input size.