The identification of combinatorial intervention strategies and the elucidation of failure modes that may cause aberrant behavior of cellular signaling networks are highly relevant topics in cell biology, medicine, and pharmaceutical industry. We have recently introduced the concept of minimal intervention sets (MISs)--minimal combinations of knock-ins and knock-outs provoking a desired/observed response in certain target nodes--to tackle those problems within a Boolean/logical framework. We first generalize the notion of MISs and then present several techniques for search space reduction facilitating the enumeration of MISs in networks of realistic size. One strategy exploits topological information about network-wide interdependencies between the nodes to discard unfavorable single interventions. A similar technique checks during the algorithm whether all target nodes of an intervention problem can be influenced in appropriate direction (up/down) by the interventions contained in MIS candidates. Another strategy takes lessons from electrical engineering: certain interventions are equivalent with respect to their effect on the target nodes and can therefore be grouped in fault equivalence classes (FECs). FECs resulting from so-called structural equivalence can be easily computed in a preprocessing step, with the advantage that only one representative per class needs to be considered when constructing the MISs in the main algorithm. With intervention problems from realistic networks as benchmarks, we show that these algorithmic improvements may reduce the computation time up to 99%, increasing the applicability of MISs in practice.