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A branch-&-price approach to the unrooted maximum agreement forest problem

Martin Frohn, Steven Kelk, Simona Vychytilova

TL;DR

This paper presents the first branch-&-price algorithm for the unrooted maximum agreement forest problem ($u$MAF) on two unrooted X-labelled binary trees, formulating an exponential-size ILP whose master problem is solved by column generation. The pricing subproblem is shown to be a weighted maximum agreement subtree problem ($\mathrm{WMAST}$), solvable in $O(n^2)$ time via a dynamic-programming approach extended to fixed weights. Experimental results on the reference dataset demonstrate that, with polynomial-time kernelization, the method solves 731 of 735 instances within 5 minutes (the remaining 4 in 5–7 minutes), outperforming a state-of-the-art FPT branching approach on many challenging cases. The study highlights that the pricing step is the main bottleneck and suggests directions for speedups, while also noting the potential to extend the framework to rooted MAF and multi-tree variants. Overall, the branch-&-price approach achieves state-of-the-art performance and provides a practical, scalable exact solver for uMAF in realistic settings.

Abstract

We propose the first branch-&-price algorithm for the maximum agreement forest problem on unrooted binary trees: given two unrooted X-labelled binary trees we seek to partition X into a minimum number of blocks such that the induced subtrees are disjoint and have the same topologies in both trees. We provide a dynamic programming algorithm for the weighted maximum agreement subtree problem to solve the pricing problem. When combined with rigorous polynomial-time pre-processing our branch-&-price algorithm exhibits (beyond) state-of-the-art performance.

A branch-&-price approach to the unrooted maximum agreement forest problem

TL;DR

This paper presents the first branch-&-price algorithm for the unrooted maximum agreement forest problem (MAF) on two unrooted X-labelled binary trees, formulating an exponential-size ILP whose master problem is solved by column generation. The pricing subproblem is shown to be a weighted maximum agreement subtree problem (), solvable in time via a dynamic-programming approach extended to fixed weights. Experimental results on the reference dataset demonstrate that, with polynomial-time kernelization, the method solves 731 of 735 instances within 5 minutes (the remaining 4 in 5–7 minutes), outperforming a state-of-the-art FPT branching approach on many challenging cases. The study highlights that the pricing step is the main bottleneck and suggests directions for speedups, while also noting the potential to extend the framework to rooted MAF and multi-tree variants. Overall, the branch-&-price approach achieves state-of-the-art performance and provides a practical, scalable exact solver for uMAF in realistic settings.

Abstract

We propose the first branch-&-price algorithm for the maximum agreement forest problem on unrooted binary trees: given two unrooted X-labelled binary trees we seek to partition X into a minimum number of blocks such that the induced subtrees are disjoint and have the same topologies in both trees. We provide a dynamic programming algorithm for the weighted maximum agreement subtree problem to solve the pricing problem. When combined with rigorous polynomial-time pre-processing our branch-&-price algorithm exhibits (beyond) state-of-the-art performance.
Paper Structure (13 sections, 1 theorem, 13 equations, 1 figure, 1 table)

This paper contains 13 sections, 1 theorem, 13 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A branch-&-price formulation for the unrooted maximum agreement forest problem
  4. The weighted maximum agreement subtree problem
  5. Implementation of the branch-&-price algorithm
  6. Computational experiments
  7. The reference dataset of Van Wersch et al.
  8. Time to generate columns, and number of columns generated
  9. Termination
  10. Branching
  11. Solving time
  12. Discussion and conclusion
  13. Acknowledgements

Key Result

Proposition 1

For $i\in\{1,2\}$, let $t_i$ be a rooted phylogenetic $X_i$-tree. Let $\rho$ be the root of the agreement subtree corresponding to $\mathcal{M}\left(t_1,t_2\right)$. Then,

Figures (1)

  • Figure 1: A branch-&-price scheme for uMAF. Intermediate solutions, the branching and pricing are shown in blue, red and yellow, respectively.

Theorems & Definitions (2)

  • Proposition 1
  • proof