Given a set of genomes g and a distance measure d, the genome rearrangement median problem asks for another genome q that minimizes Sigma n is an element of g(d(q,g)). This problem lies at the heart of phylogenetic reconstruction from rearrangement data. where Solutions to the median problems are iteratively used to update genome assignments to internal nodes for a given tree. The median problem for reversal distance and DCJ distance is known to be NP-hard. regardless of whether genomes contain circular chromosomes or linear chromosomes and of whether extra circular chromosomes is allowed in the median genomes. In this paper, we study the relaxed DCJ median problem on linear multichromosomal genomes where the median genomes may contain extra circular chromosomes. extend our prior results on circular genomes-which allowed us to compute exact medians for genomes of up to 1,000 genes within a few minutes. First we model the DCJ median problem on linear multichromosomal genomes by a cupped multiple break-point graph, a model that avoids another computationally difficult problem-a multi-way capping problem for linear genomes, then establish its corresponding decomposition theory, and finally show its results on genomes with up to several thousand genes.