The Isolation Technique

Abstract

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.

Brother: And the Lord spake, saying, “First shalt thou take out the Holy Pin. Then, shalt thou count to three, no more, no less. Three shalt be the number thou shalt count, and the number of the counting shalt be three. Four shalt thou not count, nor either count thou two, excepting that thou then proceed to three. Five is right out. Once the number three, being the third number, be reached, then lobbest thou thy Holy Hand Grenade of Antioch towards thy foe, who being naughty in my sight, shall snuff it.”

Maynard: Amen. All: Amen.

Arthur: Right! One… two…five!

—Monty Python and the Holy Grail