Characterizing Average-Case Complexity of PH by Worst-Case Meta-Complexity - Hirahara, Shuichi (Proceedings / annual Symposium on Foundations of Computer Science, 2020)
We exactly characterize the average-case complexity of the polynomial-time hierarchy (PH) by the worst-case (meta-)complexity of GapMINKT PH , i.e., an approximation version of the problem of determining if a given string can be compressed to a short PH-oracle efficient program. Specifically, we establish the following equivalence: \begin{align*} &\text{DistPH}\subseteq \text{AvgP}\ \ (\mathrm{i}.\mathrm{e}.,\ \text{PH}\ \mathrm{is\ easy\ on\ average})\\ \Longleftrightarrow &\text{GapMINKT}^{\text{PH}}\in \mathrm{P}. \end{align*} In fact, our equivalence is significantly broad: A number of statements on several fundamental notions of complexity theory, such as errorless and one-sided-error average-case complexity, sublinear-time-bounded and polynomial-time-bounded Kolmogorov complexity, and PH-computable hitting set generators, are all shown to be equivalent. Our equivalence provides fundamentally new proof techniques for analyzing average-case complexity through the lens of meta-complexity of time-bounded Kolmogorov complexity and resolves, as immediate corollaries, questions of equivalence among different notions of average-case complexity of PH: low success versus high success probabilities (i.e., a hardness amplification theorem for DistPH against uniform algorithms) and errorless versus one-sided-error average-case complexity of PH. Our results are based on a sequence of new technical results that further develops the proof techniques of the author's previous work on the non-black-box worst-case to average-case reduction and unexpected hardness results for Kolmogorov complexity (FOCS'18, CCC'20, ITCS'20, STOC'20). Among other things, we prove the following. 1) \text{GapMINKT}^{\text{NP}}\in \mathrm{P} implies \mathrm{P}=\text{BPP} . At the core of the proof is a new black-box hitting set generator construction whose reconstruction algorithm uses few random bits, which also improves the approximation quality of the non-black-box worst-case to average-case reduction without using a pseudorandom generator. 2) \text{GapMINKT}^{\text{PH}}\in \mathrm{P} implies \text{DistPH}\subseteq \text{AvgBPP}=\text{AvgP} . 3)If MINKT PH is easy on a 1/\text{poly}(n) -fraction of inputs, then \text{GapMINKT}^{\text{PH}}\in \mathrm{P} . This improves the error tolerance of the previous non-black-box worst-case to average-case reduction. The full version of this paper is available on ECCC.