Core Concepts
The computational complexity of enumerating all most parsimonious scenarios for transforming one genome into another under various genome rearrangement models is investigated.
Abstract
The paper examines the computational complexity of enumeration in certain genome rearrangement models. The key findings are:
In the Single Cut-and-Join (SCaJ) model, the Pairwise Rearrangement problem, which asks to compute the number of most parsimonious scenarios transforming one genome into another, is shown to be #P-complete under polynomial-time Turing reductions.
In the Single Cut or Join (SCoJ) model, the #Median problem, which asks to count the number of median genomes for a given set of genomes, is shown to be in the complexity class FL, improving upon the previous polynomial-time (FP) bound.
The paper first introduces various genome rearrangement models and associated computational problems. It then provides a detailed analysis of the complexity of enumerating sorting scenarios in the SCaJ model, establishing the #P-completeness of the Pairwise Rearrangement problem. This involves a reduction from the Multiset-Equal-Partition problem, which is shown to be #P-complete.
For the SCoJ model, the paper presents an improved upper bound on the complexity of the #Median problem, showing it belongs to the complexity class FL. This improves upon the previous FP bound.
The paper also discusses related work on efficient computational approaches, such as sampling and approximation, to cope with the intractability of enumeration in genome rearrangement problems. It highlights the close connection between approximate counting and sampling, and the challenges in developing efficient uniform or near-uniform samplers.
Stats
The sum of the sizes of all crowns in the adjacency graph is 2p - 2n, where p is an odd prime and n is a positive integer.
The number of crowns in the adjacency graph is 2n + 2.
Quotes
"Genome rearrangement models consider situations in which large scale mutations alter the order of the genes within the genome."
"Subsequent to his work on Drosophila, Sturtevant together with Novitski [55] introduced one of the first genome rearrangement problems, seeking a minimum length sequence of operations (in particular, so-called reversals [26]) that would transform one genome into another."
"When choosing an appropriate model, it is important to balance biological relevance with computational tractability. This motivates the study of the computational complexity for genome rearrangement problems."