Identifying Breakpoint Median Genomes: A Branching Algorithm Approach

Genome comparison often involves quantifying dissimilarities between genomes with identical gene sets, commonly using breakpoints - points where adjacent genes in one genome are not adjacent in another. The concept of a median genome, used for comparison of multiple genomes, aims to find a genome that minimizes the total distance to all genomes in a given set. While median genomes are useful for extracting common genomic information and estimating ancestral traits, the existence of multiple divergent medians raises concerns about their accuracy in reflecting the true ancestor. The median problem is known to be NP-hard, particularly for unichromosomal genomes, and solving it becomes increasingly challenging under different genome distance models. In this work, we introduce a novel branching algorithm to efficiently find all breakpoint medians of k linear unichromosomal genomes, represented as unsigned permutations. This algorithm constructs a rooted labeled tree, where the sequence of labels along each complete ray defines a genome, providing a structured and efficient way to explore the space of candidate medians by narrowing the search to a well-defined and significantly smaller subset of the permutation space. We validate our approach with experiments on randomly generated sets of three permutations. The results show that our method successfully finds the exact medians and also identifies many near-optimal approximations. Our experiments further show that most medians lie relatively close to the input permutations, in agreement with prior theoretical results.