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Motivated by the trend of genome sequencing without completing the sequence of the whole genomes, a problem on filling an incomplete multichromosomal genome (or scaffold) I with respect to a complete target genome G was studied. The objective is to minimize the resulting genomic distance between I' and G, where I' is the corresponding filled scaffold. We call this problem the one-sided scaffold filling problem. In this paper, we conduct a systematic study for the scaffold filling problem under the breakpoint distance and its variants, for both unichromosomal and multichromosomal genomes (with and without gene repetitions). When the input genome contains no gene repetition (i.e., is a fragment of a permutation), we show that the two-sided scaffold filling problem (i.e., G is also incomplete) is polynomially solvable for unichromosomal genomes under the breakpoint distance and for multichromosomal genomes under the genomic (or DCJ-Double-Cut-and-Join) distance. However, when the input genome contains some repeated genes, even the one-sided scaffold filling problem becomes NP-complete when the similarity measure is the maximum number of adjacencies between two sequences. For this problem, we also present efficient constant-factor approximation algorithms: factor-2 for the general case and factor 1.33 for the one-sided case.