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

2024-02-02

Scandinavian Workshop on Algorithm Theory

AI Summary

This paper addresses the pairwise genome rearrangement counting problem under the Single-Cut-or-Join (SCJ) model—i.e., exactly counting the number of shortest SCJ sequences transforming one genome into another. This problem is known to be #P-complete, rendering exact counting and uniform sampling intractable in general. We establish its first parameterized tractability: we design the first fixed-parameter tractable (FPT) algorithm parameterized by the number (k) of nontrivial connected components in the adjacency graph, achieving runtime (f(k) cdot ext{poly}(n)). As a corollary, the problem is also FPT with respect to the rearrangement distance (d). Our work breaks through the #P-completeness barrier and identifies the number of nontrivial components as the key structural parameter governing computational efficiency. This yields the first efficient, exact counting tool for SCJ-based genome evolution modeling and statistical inference.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Models for Web evolution

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 $# extsf{P}$-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.

Problem

• Study Pairwise Rearrangement problem in genome evolution
• Address intractability via parameterized complexity analysis
• Develop FPT algorithm based on adjacency graph components

Innovation

• Fixed-parameter tractable algorithm for Pairwise Rearrangement
• Parameterized complexity based on adjacency graph components
• Efficient sampling via non-trivial component count