The automata-theoretic approach to the problem of program verification requires efficient minimization and complementation of nondeterministic finite automata. This work presents a direct empirical comparison of well-known automata minimization algorithms, and also of a symbolic and an explicit approach to complementing automata. I propose a probabilistic framework for testing the performance of automata-theoretic algorithms, and use it to compare empirically Brzozowski's and Hopcroft's minimization algorithms. While Hopcroft's algorithm has better overall performance, the experimental results show that Brzozowski's algorithm performs better for "high-density" automata. In this work I also analyze complementation by considering automaton universality as a model-checking problem. A novel encoding presented here allows this problem to be solved symbolically via a model-checker. I compare the performance of this approach to that of the standard explicit algorithm which is based on the subset construction, and show that the explicit approach unexpectedly performs an order of magnitude better.