One of the most important processing steps in any analysis pipeline is handling missing data. Traditional approaches simply delete any sample or feature with missing elements. Recent imputation methods replace missing data based on assumed relationships between observed data and the missing elements. However, there is a largely under-explored alternative amid these extremes. Partial deletion approaches remove excessive amounts of missing data, as defined by the user. They can be used in place of traditional deletion or as a precursor to imputation. In this manuscript, we expand upon the Mr. Clean suite of algorithms, focusing on the scenario where all missing data is removed. We show that the RowCol Integer Program can be recast as a Linear Program, thereby reducing runtime. Additionally, the Element Integer Program can be reformulated to reduce the number of variables and allow for high levels of parallelization. Using real-world data sets from genetic, gene expression, and single cell RNA-seq experiments we demonstrate that our algorithms outperform existing deletion techniques over several missingness values, balancing runtime and data retention. Our combined greedy algorithm retains the maximum number of valid elements in 126 of 150 scenarios and stays within 1\% of maximum in 23 of the remaining experiments. The reformulated Element IP complements the greedy algorithm when removing all missing data, boasting a reduced runtime and increase in valid elements in larger data sets, over its generic counterpart. These two programs greatly increase the amount of valid data retained over traditional deletion techniques and further improve on existing partial deletion algorithms.