Pairwise Rearrangement is Fixed-Parameter Tractable in the Single Cut-and-Join Model

TL;DR

This paper studies the Pairwise Rearrangement problem in the SCaJ model, where exact counting is #P-complete. The authors develop a combinatorial dynamic-programming framework that leverages the adjacency graph’s component types (N, W, M, and crowns) to count sorting scenarios via partitions and to compute the number of most parsimonious sequences. They prove that the problem is fixed-parameter tractable by the number $k$ of nontrivial components in the adjacency graph and provide a two-stage DP to build a lookup table of subdivision counts, plus a partition-based aggregation to obtain $ ext{ #MPS}$. They further show that uniform sampling of sorting scenarios is also fixed-parameter tractable by the same parameter, enabling practical sampling in plausible biological regimes. Collectively, these results identify the number of nontrivial adjacency-graph components as the key obstacle to efficient sampling and enumeration, and they offer a concrete, implementable approach for both counting and sampling within FPT bounds.

Abstract

Genome rearrangement is a common model for molecular evolution. In this paper, we consider the Pairwise Rearrangement problem, which takes as input two genomes and asks for the number of minimum-length sequences of permissible operations transforming the first genome into the second. In the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010), Pairwise Rearrangement is $\#\textsf{P}$-complete (Bailey, et. al., COCOON 2023), which implies that exact sampling is intractable. In order to cope with this intractability, we investigate the parameterized complexity of this problem. We exhibit a fixed-parameter tractable algorithm with respect to the number of components in the adjacency graph that are not cycles of length $2$ or paths of length $1$. As a consequence, we obtain that Pairwise Rearrangement in the Single Cut-and-Join model is fixed-parameter tractable by distance. Our results suggest that the number of nontrivial components in the adjacency graph serves as the key obstacle for efficient sampling.

Figures (4)

• Figure 1: An edge-labeled genome bailey2023complexity.
• Figure 2: (i) Adjacency $X_2^hX_3^t$ is cut. (ii) Telomeres $X_1^h$ and $X_3^h$ are joined. (iii) Adjacency $X_2^hX_3^t$ is cut, and resulting telomere $X_2^h$ is joined with $X_1^h$. (iv) Adjacencies $X_1^tX_2^t$ and $X_2^hX_3^t$ are replaced with $X_1^tX_2^h$ and $X_2^tX_3^t$bailey2023complexity
• Figure 3: The adjacency graph $A(G_1,G_2)$ is shown in the middle, for genomes $G_1$ and $G_2$ shown above and below, respectively bailey2023complexity.
• Figure 4: This figure depicts the operations described in Observation \ref{['obs:SortingScenarios']}, where the arrows point from the original component type(s) to the component type(s) produced by the three operations allowed in the SCaJ model: cut, join, and cut-join. Here, $T$ denotes a trivial crown and $C$ denotes a nontrivial crown. The eight diagrams in (I) show each of the sorting operations (a)--(h) where bold single arrows represent cut-join operations, dashed arrows represent cut operations, and the double arrow represents the join operation. Diagram (II) summarizes which components can be produced. Note that all operations will only result in $W$-shaped components, $N$-shaped components, and/or trivial crowns.

Theorems & Definitions (39)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.8: bailey2023complexity