On the computational complexity of the maximum parsimony reconciliation problem in the duplication-loss-coalescence model

Abstract

BackgroundPhylogenetic tree reconciliation is a widely-used method for inferring the evolutionary histories of genes and species. In the duplication-loss-coalescence (DLC) model, we seek a reconciliation that explains the incongruence between a gene and species tree using gene duplication, loss, and deep coalescence events. In the maximum parsimony framework, costs are associated with these event types and a reconciliation is sought that minimizes the total cost of the events required to map the gene tree onto the species tree.ResultsWe show that this problem is NP-hard even for the special case of minimizing the number of duplications. We then show that the problem is APX-hard when both duplications and losses are considered, implying that no polynomial-time approximation scheme can exist for the problem unless P = NP.ConclusionsThese intractability results are likely to guide future research on algorithmic aspects of the DLC-reconciliation problem.

An Integer Linear Programming Solution for the Most Parsimonious Reconciliation Problem under the Duplication-Loss-Coalescence Model

“…For convenience, rather than specify the locus map directly, we specify the placement of duplications along gene branches. Via a trivial extension of Theorem 1 of Bork et al [1], these two characterizations are equivalent, and thus, we can use locus maps and duplication placements interchangeably. By requiring a duplication to create a new locus, a duplication placement automatically satisfies LCT constraint 3.…”

“…However, the MPR problem for the DLC model is NP-hard and even hard to approximate (APX-hard) [1]. While the current dynamic programming algorithm provides an exact solution and is fixed-parameter tractable (when parameterized by the number of genes that map to any given species) 1 [7], it has worst-case exponential runtime and can be problematic to use in practice due to time and memory requirements.…”

“…Furthermore, users can sacrifice accuracy by limiting the maximum runtime of the ILP solver, making it an attractive choice for phylogenetic pipelines. 1 Let m denote the number of leaves in the species tree, n denote the number of leaves in the gene tree, and k denote the maximum number of nodes at the top or bottom of any species branch. The value of k is induced by the LCA species map.…”

Multiple Optimal Reconciliations Under the Duplication-Loss-Coalescence Model

“…We previously showed that the MPR problem for the DLC model is NP-complete and even hard to approximate (APX-complete), and it is therefore unlikely that polynomialtime algorithms or approximation schemes exist for this problem [3]. Thus, unsurprisingly, the DLCpar algorithm has worst-case exponential runtime.…”

The Most Parsimonious Reconciliation Problem in the Presence of Incomplete Lineage Sorting and Hybridization is NP-Hard

“…When considering duplications, transfers, and losses, the MPR problem can be solved in polynomial time or is NP-hard depending on whether the species tree is undated, partially dated, or fully dated, and on whether the reconciliation is constrained to be time-consistent (Ovadia et al 2011, Tofigh et al 2011). Similarly, depending on details of the underlying model, the MPR problem can be solved in polynomial time when considering duplications, transfers, losses, and coalescence (Stolzer et al 2012, Chan et al 2017), or is NP-hard when considering duplications, losses, and coalescence (Bork et al 2017). The MPR problem is also NP-hard when simultaneously modeling the evolution of domains, genes, and species (Li and Bansal 2019).…”