License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.CCC.2022.35
URN: urn:nbn:de:0030-drops-165975
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Liu, Yanyi ; Pass, Rafael

Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity

LIPIcs-CCC-2022-35.pdf (0.7 MB)


A central open problem in complexity theory concerns the question of whether all efficient randomized algorithms can be simulated by efficient deterministic algorithms. We consider this problem in the context of promise problems (i.e,. the prBPP v.s. prP problem) and show that for all sufficiently large constants c, the following are equivalent:
- prBPP = prP.
- For every BPTIME(n^c) algorithm M, and every sufficiently long z ∈ {0,1}ⁿ, there exists some x ∈ {0,1}ⁿ such that M fails to decide whether Kt(x∣z) is "very large" (≥ n-1) or "very small" (≤ O(log n)). where Kt(x∣z) denotes the Levin-Kolmogorov complexity of x conditioned on z. As far as we are aware, this yields the first full characterization of when prBPP = prP through the hardness of some class of problems. Previous hardness assumptions used for derandomization only provide a one-sided implication.

BibTeX - Entry

  author =	{Liu, Yanyi and Pass, Rafael},
  title =	{{Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{35:1--35:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-165975},
  doi =		{10.4230/LIPIcs.CCC.2022.35},
  annote =	{Keywords: Derandomization, Kolmogorov Complexity, Hitting Set Generators}

Keywords: Derandomization, Kolmogorov Complexity, Hitting Set Generators
Collection: 37th Computational Complexity Conference (CCC 2022)
Issue Date: 2022
Date of publication: 11.07.2022

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