Genetic recombination plays two essential biological roles. It ensures the fidelity of the transmission of genetic information from one generation to the next and it generates new combinations of genetic variants. Therefore, recombination is a critical process in shaping arrangement of polymorphisms within populations. "Recombination breakpoints" in a given set of genomes from individuals in a population divide the genome into haplotype blocks, resulting in a mosaic structure on the genome. In this paper, we study the Minimum Mosaic Problem: given a set of genome sequences from individuals within a population, compute a mosaic structure containing the minimum number of breakpoints. This mosaic structure provides a good estimation of the minimum number of recombination events (and their location) required to generate the existing haplotypes in the population. We solve this problem by finding the shortest path in a directed graph. Our algorithm's efficiency permits genome-wide analysis.