Ryan Williams
ABSTRACT
The P vs NP problem arose from the question of whether exhaustive search is necessary for problems with short verifiable solutions. We still do not know if even a slight algorithmic improvement over exhaustive search is universally possible for all NP problems, and to date no major consequences have been derived from the assumption that an improvement exists.
We show that there are natural NP and BPP problems for which minor algorithmic improvements over the trivial deterministic simulation already entail lower bounds such as NEXP is not in P/poly and LOGSPACE is not equal to NP. These results are especially interesting given that similar improvements have been found for many other hard problems. Optimistically, one might hope our results suggest a new path to lower bounds; pessimistically, they show that carrying out the seemingly modest program of finding slightly better algorithms for all search problems may be extremely difficult (if not impossible).
We also prove unconditional superpolynomial time-space lower bounds for improving on exhaustive search.
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- Improving exhaustive search implies superpolynomial lower bounds
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