We revisit the question of whether worst-case hardness of the time-bounded Kolmogorov complexity problem, MINK^poly - that is, determining whether a string is "structured" (i. e. , Kᵗ (x) n - log n) - characterizes OWF, but with either of the following caveats (1) considering a non-standard notion of probabilistic Kᵗ, as opposed to the standard notion of Kᵗ, or (2) assuming somewhat strong, and non-standard, derandomization assumptions. In this paper, we present an alternative method for establishing their result which enables significantly weakening the caveats. First, we show that boundary hardness of the more standard randomized Kᵗ problem suffices (where randomized Kᵗ (x) is defined just like Kᵗ (x) except that the program generating the string x may be randomized). As a consequence of this result, we can provide a characterization also in terms of just "plain" Kᵗ under the most standard derandomization assumption (used to derandomize just BPP into P) - namely E ̸ ⊆ ioSIZE2^o (n). Our proof relies on language compression schemes of Goldberg-Sipser (STOC'85) ; using the same technique, we also present the the first worst-case to average-case reduction for the exact MINK^poly problem (under the same standard derandomization assumption), improving upon Hirahara’s celebrated results (STOC'18, STOC'21) that only applied to a gap version of the MINK^poly problem, referred to as GapMINK^poly, where the goal is to decide whether Kᵗ (x) ≤ n-O (log n) ) or K^poly (t) (x) ≥ n-1 and under the same derandomization assumption.
Liu et al. (Thu,) studied this question.
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