On sampling SCJ rearrangement scenarios

TL;DR

This paper investigates the computational complexity of sampling and counting most parsimonious SCJ genome rearrangement scenarios, revealing polynomial-time solutions for two genomes but proving hardness for multiple genomes, with broader implications for discrete character evolution.

Contribution

It establishes the complexity boundaries of sampling and counting SCJ scenarios, showing polynomial-time solutions for two genomes and hardness results for multiple genomes, connecting to broader discrete character evolution problems.

Findings

Sampling and counting are easy for two genomes.

Counting and sampling are hard for multiple genomes.

Hardness results imply RP ≠ NP unless certain algorithms exist.

Abstract

The Single Cut or Join (SCJ) operation on genomes, generalizing chromosome evolution by fusions and fissions, is the computationally simplest known model of genome rearrangement. While most genome rearrangement problems are already hard when comparing three genomes, it is possible to compute in polynomial time a most parsimonious SCJ scenario for an arbitrary number of genomes related by a binary phylogenetic tree. Here we consider the problems of sampling and counting the most parsimonious SCJ scenarios. We show that both the sampling and counting problems are easy for two genomes, and we relate SCJ scenarios to alternating permutations. However, for an arbitrary number of genomes related by a binary phylogenetic tree, the counting and sampling problems become hard. We prove that if a Fully Polynomial Randomized Approximation Scheme or a Fully Polynomial Almost Uniform Sampler exist for the most parsimonious SCJ scenario, then RP = NP. The proof has a wider scope than genome rearrangements: the same result holds for parsimonious evolutionary scenarios on any set of discrete characters.