Many recognition systems use constrained search to locate objects in cluttered environments. Earlier analysis showed that the expected search is quadratic in the number of model and data features, if all the data comes from one object, but is exponential when spurious data is included. To overcome this, many methods terminate search once an interpretation that is "good enough" is found. We formally examine the combinatorics of this, showing that correct termination procedures dramatically reduce search. We provide conditions on the object model and the scene clutter such that the expected search is quartic. These results are shown to agree with empirical data for cluttered object recognition.