Reflections on kernelizing and computing unrooted agreement forests

Abstract

Phylogenetic trees are leaf-labelled trees used to model the evolution of species. Here we explore the practical impact of kernelization (i.e. data reduction) on the NP-hard problem of computing the TBR distance between two unrooted binary phylogenetic trees. This problem is better-known in the literature as the maximum agreement forest problem, where the goal is to partition the two trees into a minimum number of common, non-overlapping subtrees. We have implemented two well-known reduction rules, the subtree and chain reduction, and five more recent, theoretically stronger reduction rules, and compare the reduction achieved with and without the stronger rules. We find that the new rules yield smaller reduced instances and thus have clear practical added value. In many cases they also cause the TBR distance to decrease in a controlled fashion, which can further facilitate solving the problem in practice. Next, we compare the achieved reduction to the known worst-case theoretical bounds of $$15k-9$$ 15 k - 9 and $$11k-9$$ 11 k - 9 respectively, on the number of leaves of the two reduced trees, where k is the TBR distance, observing in both cases a far larger reduction in practice. As a by-product of our experimental framework we obtain a number of new insights into the actual computation of TBR distance. We find, for example, that very strong lower bounds on TBR distance can be obtained efficiently by randomly sampling certain carefully constructed partitions of the leaf labels, and identify instances which seem particularly challenging to solve exactly. The reduction rules have been implemented within our new solver Tubro which combines kernelization with an Integer Linear Programming (ILP) approach. Tubro also incorporates a number of additional features, such as a cluster reduction and a practical upper-bounding heuristic, and it can leverage combinatorial insights emerging from the proofs of correctness of the reduction rules to simplify the ILP.

Sharp Upper and Lower Bounds on a Restricted Class of Convex Characters

“…The random trees were generated following the same protocol as used in in [16], which we summarize here. Starting with an empty rooted binary phylogenetic tree, we place a taxon from X on one side of the root with probability 1 2 , and on the other side with probability 1 2 , and then recurse on the two sides of the root until the leaves are reached.…”

“…This was noted in [12] where an algorithm based on listing g 2 characters was used to design a simple but surprisingly practical algorithm for exact computation of maximum parsimony distance [6] phylogenetic trees. This later became the foundation for a far more scalable samplingbased heuristic for the same problem [16]; both the listing and sampling leverage the dynamic programming scaffolding originally used to actually count g k . In [12] it was also observed that the maximum agreement forest problem [2] can be solved in time O(2.619 n ) by listing g 1 characters.…”

A Near-Linear Kernel for Two-Parsimony Distance

“…54(lg n+1) , as claimed in Theorem 13. To do this, we use an ILP formulation of the unrooted MAF problem by Van Wersch et al [13]. For a pair of trees (T 1 , T 2 ) on X , let Q be the set of incompatible quartets of T 1 and T 2 .…”

“…Such a set of quartets does not give a parsimony distance that is at least the number of quartets, but the parsimony distance is still linear in the number of quartets. We present a primal-dual algorithm based on an ILP formulation of the maximum agreement forest problem [13] that finds such a set Q of incompatible quartets and an AF of size O(|Q| • lg |X |). This establishes the key claim that…”