On the Complexity of the (𝓁, k)-Median Problems

Abstract

The genome median problem is a central computational problem in comparative genomics, as it models the reconstruction of an ancestral genome from a set of related genomes. Given 𝓁 genomes and a distance measure, the problem asks for a genome that minimizes the sum of the distances to the input genomes. Two classical distances are the breakpoint distance and the double-cut-and-join (DCJ) distance. For multichromosomal circular genomes, the median problem is polynomial-time solvable under the breakpoint distance, whereas it is NP-hard under the DCJ distance.
For even integer k ≥ 2, the σ_k distance interpolates between these two extremes: σ₂ corresponds to the breakpoint distance, while σ_∞ corresponds to the DCJ distance. A central open problem in this setting is the (3,4)-Median problem, which asks for a median of three genomes under the σ₄ distance, the first intermediate distance after the breakpoint distance.
Motivated by this question, we study the more general (𝓁,k)-Median problem, in which 𝓁 is the number of input genomes and k determines the σ_k distance. We prove that (𝓁,6)-Median for every 𝓁 ≥ 4 and (3,12)-Median are NP-complete. We then extend the hardness of (𝓁,6)-Median to (𝓁,k)-Median for all even k ≥ 6, and the hardness of (3,12)-Median to (3,k)-Median for all even k ≥ 12. These results identify broad hardness regions in the (𝓁,k) parameter space and delimit the remaining open cases around the fundamental (3,4)-Median problem.