Read trimming is a fundamental first step of the analysis of next generation sequencing (NGS) data. Traditionally, it is performed heuristically, and algorithmic work in this area has been neglected. Here, we address this topic and formulate three optimization problems for block-based trimming (truncating the same low-quality positions at both ends for all reads and removing low-quality truncated reads). We find that all problems are NP-hard. Hence, we investigate the approximability of the problems. Two of them are NP-hard to approximate. However, the non-random distribution of quality scores in NGS data sets makes it tempting to speculate that quality constraints for read positions are typically satisfied by fulfilling quality constraints for reads. Thus, we propose three relaxed problems and develop efficient polynomial-time algorithms for them including heuristic speed-up techniques and parallelizations. We apply these optimized block trimming algorithms to 12 data sets from three species, four sequencers, and read lengths ranging from 36 to 101 bp and find that (i) the omitted constraints are indeed almost always satisfied, (ii) the optimized read trimming algorithms typically yield a higher number of untrimmed bases than traditional heuristics, and (iii) these results can be generalized to alternative objective functions beyond counting the number of untrimmed bases.