2026-07-26T00:33:35Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:21044 2024-09-16T06:02:39Z ddc:004 open_access

Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity Goldberg, Halley Kabanets, Valentine Meta-complexity Randomized reductions NP-hardness Worst-case complexity Time-bounded Kolmogorov complexity A central open question within meta-complexity is that of NP-hardness of problems such as MCSP and MK^{t}P. Despite a large body of work giving consequences of and barriers for NP-hardness of these problems under (restricted) deterministic reductions, very little is known in the setting of randomized reductions. In this work, we give consequences of randomized NP-hardness reductions for both approximating and exactly computing time-bounded and time-unbounded Kolmogorov complexity. In the setting of approximate K^{poly} complexity, our results are as follows. 1) Under a derandomization assumption, for any constant δ > 0, if approximating K^t complexity within n^{δ} additive error is hard for SAT under an honest randomized non-adaptive Turing reduction running in time polynomially less than t, then NP = coNP. 2) Under the same assumptions, the worst-case hardness of NP is equivalent to the existence of one-way functions. Item 1 above may be compared with a recent work of Saks and Santhanam [Michael E. Saks and Rahul Santhanam, 2022], which makes the same assumptions except with ω(log n) additive error, obtaining the conclusion NE = coNE. In the setting of exact K^{poly} complexity, where the barriers of Item 1 and [Michael E. Saks and Rahul Santhanam, 2022] do not apply, we show: 3) If computing K^t complexity is hard for SAT under reductions as in Item 1, then the average-case hardness of NP is equivalent to the existence of one-way functions. That is, "Pessiland" is excluded. Finally, we give consequences of NP-hardness of exact time-unbounded Kolmogorov complexity under randomized reductions. 4) If computing Kolmogorov complexity is hard for SAT under a randomized many-one reduction running in time t_R and with failure probability at most 1/(t_R)^16, then coNP is contained in non-interactive statistical zero-knowledge; thus NP ⊆ coAM. Also, the worst-case hardness of NP is equivalent to the existence of one-way functions. We further exploit the connection to NISZK along with a previous work of Allender et al. [Eric Allender et al., 2023] to show that hardness of K complexity under randomized many-one reductions is highly robust with respect to failure probability, approximation error, output length, and threshold parameter. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Halley Goldberg and Valentine Kabanets 2024 Is Part Of LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2024.51 urn:nbn:de:0030-drops-210444 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.51 eng https://creativecommons.org/licenses/by/4.0/legalcode