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Many interesting combinatorial problems were found to be NP-complete. Since there is little hope to solve them fast in the worst case, researchers look for algorithms which are fast just “on average”. This matter is sensitive to the choice of a particular NP-complete problem and a probability distribution of its instances. Some of these tasks were easy and some not. But one needs a way to distinguish the “difficult on average” problems. Such negative results could not only save “positive” efforts but may also be used in areas (like cryptography) where hardness of some problems is a frequent assumption. It is shown below that the Tiling problem with uniform distribution of instances has no polynomial “on average” algorithm, unless every NP-problem with every simple probability distribution has it. It is interesting to try to prove similar statements for other NP-problems which resisted so far “average case” attacks.
Leonid A. Levin (Sat,) studied this question.