The Complexity and Distribution of Hard Problems

A weak completeness phenomenon is investigated in the complexity class ${\rm E}= {\rm DTIME}(2^{\rm linear})$. According to standard terminology, a language $H$ is $\leq^{\rm P}_{m}$- hard for E if the set ${\rm P}_{m}(H)$, consisting of all languages ...

The main theorem of this paper is that, for every real number $\alpha<1$ (e.g., $\alpha=0.99$), only a measure 0 subset of the languages decidable in exponential time are $\leq^{P}_{n^{\alpha} - tt}$-reducible to languages that are not exponentially ...

Given a real number $\alpha < 1$, every language that is weakly $\leq_{n^{\alpha / 2} - {\rm T}}^{{\rm P}} $-hard for E or weakly $\leq_{n^{\alpha} - {\rm T}}^{\rm P}$-hard for E₂ is shown to be exponentially dense. This simultaneously strengthens the ...