The Gap Between Average-Case and Worst-Case Complexity

Abstract

Understanding the average-case complexity of $\mathsf{NP}$ is a central goal in complexity theory and cryptography. A necessary milestone is to understand the corresponding average-case versus worst-case question for the polynomial hierarchy ($\mathsf{PH}$). In a celebrated result, Hirahara [Hir21] proved that if every distributional problem in $\mathsf{PH}$ is easy on average, then every language in $\mathsf{PH}$ admits worst-case algorithms running in time $2^{O(n/\log n)}$. Remarkably, although the statement of Hirahara's theorem makes no reference to meta-complexity, all known proofs appear to use time-bounded Kolmogorov complexity and its associated meta-computational problems in an essential way. We show that the reliance on techniques and perspectives from meta-complexity is not simply a coincidence. Building on previous work of Hirahara [Hir20] and on a recent result of Hu, Lu, and Oliveira [HLO25], we prove that relating the average-case and worst-case complexities of $\mathsf{PH}$ is in fact equivalent to a purely meta-computational implication: efficient algorithms for a standard gap version of $\mathsf{MINpKT}$ with $\mathsf{PH}$ oracle access imply efficient algorithms for a mild gap version. Crucially, in these promise problems the standard gap separates a YES instance with time bound $t$ from a NO instance with time bound $\mathsf{poly}(t)$, whereas the mild gap keeps the YES threshold $t$ and moves the NO threshold to $t+\mathsf{poly}(n)$. Together with prior work, this equivalence points to the study of gap reductions as a potentially promising route toward understanding the relation between average-case and worst-case complexity. We make progress by proving that the gap in running time can be eliminated in the average-case setting. Finally, we prove that in the worst-case setting any efficient reduction between the two problems must use nonrelativizing techniques.