Block Interchange and Reversal Distance on Unbalanced Genomes

One method for inferring the evolutionary distance between two organisms is to find the rearrangement distance, which is defined as the minimum number of genome rearrangements required to transform one genome into the other. Rearrangements that do not alter the genome content are known as conservative. Examples of such rearrangements include: reversal, which reverts a segment of the genome; transposition, which exchanges two consecutive blocks; block interchange (BI), which exchanges two blocks at any position in the genome; and double cut and join (DCJ), which cuts two different pairs of adjacent blocks and joins them in a different manner. Initially, works in this area involved comparing genomes that shared the same set of conserved blocks. Nowadays, researchers are investigating unbalanced genomes (genomes with a distinct set of genes), which requires the use of non-conservative rearrangements such as insertions and deletions (indels). In cases where there are no repeated blocks and the genomes have the same set of blocks, the BI Distance and the Reversal Distance have polynomial-time algorithms, while the complexity of the BI and Reversal Distance problem remains unknown. In this study, we investigate the BI and Indel Distance and the BI, Reversal, and Indel Distance on genomes with different gene content and no repeated genes. We present 2-approximation algorithms for each problem using a variant of the breakpoint graph structure.