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When analyzing communities of microorganisms from their sequenced DNA, an important task is taxonomic profiling: enumerating the presence and relative abundance of all organisms, or merely of all taxa, contained in the sample. This task can be tackled via compressive-sensing-based approaches, which favor communities featuring the fewest organisms among those consistent with the observed DNA data. Despite their successes, these parsimonious approaches sometimes conflict with biological realism by overlooking organism similarities. Here, we leverage a recently developed notion of biological diversity that simultaneously accounts for organism similarities and retains the optimization strategy underlying compressive-sensing-based approaches. We demonstrate that minimizing biological diversity still produces sparse taxonomic profiles and we experimentally validate superiority to existing compressive-sensing-based approaches. Despite showing that the objective function is almost never convex and often concave, generally yielding NP-hard problems, we exhibit ways of representing organism similarities for which minimizing diversity can be performed via a sequence of linear programs guaranteed to decrease diversity. Better yet, when biological similarity is quantified by $k$-mer co-occurrence (a popular notion in bioinformatics), minimizing diversity actually reduces to one linear program that can utilize multiple $k$-mer sizes to enhance performance. In proof-of-concept experiments, we verify that the latter procedure can lead to significant gains when taxonomically profiling a metagenomic sample, both in terms of reconstruction accuracy and computational performance. Reproducible code is available at https://github.com/dkoslicki/MinimizeBiologicalDiversity.