Resolves YES if it is theoretically possible to solve all NP problems in worst-case polynomial time, with error probability bounded by a constant. Resolves NO once that is known to be impossible.

If you know a bit about complexity theory: this means it is not necessary that P = NP. BPP = NP (randomized algorithms with constant error probability) would be sufficient, and BQP = NP (quantum algorithms) would also be sufficient if the relevant algorithms could actually be run practically on quantum computers. 

 could also be sufficient if they meet the resolution criteria.

Any method of computation is allowed for this question, as long as it could in theory be physically implimented in the real universe.