The Isolation Technique
Counting is cumbersome and sometimes painful. Studying NP would indeed be far simpler if all NP languages were recognized by NP machines having at most one accepting computation path, that is, if NP = UP. The question of whether NP = UP is a nagging open issue in complexity theory. There is evidence that standard proof techniques can settle this question neither affirmatively nor negatively. However, surprisingly, with the aid of randomness we will relate NP to the problem of detecting unique solutions. In particular, we can reduce, with high probability, the entire collection of accepting computation paths of an NP machine to a single path, provided that initially there is at least one accepting computation path. We call such a reduction method an isolation technique.
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