Read trimming is a fundamental first step of the analysis of next generation sequencing (NGS) data. Traditionally, read trimming is performed heuristically, and algorithmic work in this area has been neglected. Here, we address this topic and formulate three constrained optimization problems for block-based trimming, i.e., truncating the same low-quality positions at both ends for all reads and removing low-quality truncated reads. We find that the three problems are NP-hard. 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. Based on this speculation, we propose three relaxed problems and develop efficient polynomial-time algorithms for them. We find that (i) the omitted constraints are indeed almost always satisfied and (ii) the algorithms for the relaxed problems typically yield a higher number of untrimmed bases than traditional heuristics.