eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-07-11
35:1
35:17
10.4230/LIPIcs.CCC.2022.35
article
Characterizing Derandomization Through Hardness of Levin-Kolmogorov Complexity
Liu, Yanyi
1
Pass, Rafael
1
2
Cornell Tech, New York, NY, USA
Tel-Aviv University, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol234-ccc2022/LIPIcs.CCC.2022.35/LIPIcs.CCC.2022.35.pdf
Derandomization
Kolmogorov Complexity
Hitting Set Generators