In this work we revisit computational aspects of strongly admissible semantics in Dung’s abstract argumentation frameworks. First, we complement the existing complexity analysis by focusing on the problem of computing strongly admissible sets of minimum size that contain a given argument and providing NP-hardness as well as hardness of approximation results. Based on these results, we then investigate two approaches to compute (minimum-sized) strongly admissible sets based on Answer Set Programming (ASP) and Integer Linear Programming (ILP), and provide an experimental comparison of their performance.