We propose a new approach to compute counterexamples for violated ω-regular properties of discrete-time Markov chains. Whereas most approaches compute a set of system paths as a counterexample, we determine a critical subsystem that already violates the given property. In earlier work methods have been introduced to compute such subsystems for safety properties, based on a search for shortest paths. In this paper we use mixed integer linear programming to determine minimal critical subsystems for arbitrary ω-regular properties.