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• W3165959018 abstract "We investigate the complexity of explicit construction problems, where the goal is to produce a particular object of size $n$ possessing some pseudorandom property in time polynomial in $n$. We give overwhelming evidence that ${bf APEPP}$, defined originally by Kleinberg et al., is the natural complexity class associated with explicit constructions for objects whose existence follows from the probabilistic method, by proving that a host of well-studied explicit construction problems lie in this class. We then observe that a result of Jeřabek on provability in Bounded Arithmetic, when reinterpreted as a reduction between search problems, shows that constructing a truth table of high circuit complexity is complete for ${bf APEPP}$ under ${bf P}^{bf NP}$ reductions. This demonstrates that constructing a hard truth table is a universal explicit construction problem in a concrete sense. This result in fact gives a precise algorithmic characterization of proving $2^{Omega(n)}$ circuit lower bounds for ${bf E}^{bf NP}$: the complete problem for ${bf APEPP}$ has a ${bf P}^{bf NP}$ algorithm if and only if such a lower bound holds. Together with our proof that pseudorandom generators can be constructed in ${bf APEPP}$, this also yields a self-contained and significantly simplified proof of the celebrated result of Impagliazzo and Wigderson that worst-case-hard truth tables can be used to derandomize algorithms (although the conclusion is weaker as our derandomization requires an ${bf NP}$ oracle). As another corollary of this completeness result, we show that ${bf E}^{bf NP}$ contains a language of circuit complexity $2^{Omega(n)}$ if and only if it contains a language of circuit complexity $frac{2^n}{3n}$. Finally, for several of the problems shown to lie in ${bf APEPP}$, we demonstrate direct polynomial time reductions to the explicit construction of hard truth tables." @default.
• W3165959018 created "2021-06-22" @default.
• W3165959018 creator A5045520082 @default.
• W3165959018 date "2021-06-02" @default.
• W3165959018 modified "2023-09-27" @default.
• W3165959018 title "The Hardest Explicit Construction." @default.
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• W3165959018 hasPublicationYear "2021" @default.
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