Результаты исследований: Публикации в книгах, отчётах, сборниках, трудах конференций › статья в сборнике материалов конференции › Рецензирование
Complexity of distributions and average-case hardness. / Itsykson, Dmitry; Knop, Alexander; Sokolov, Dmitry.
27th International Symposium on Algorithms and Computation, ISAAC 2016. ред. / Seok-Hee Hong. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2016. стр. 38.1-38.12 (Leibniz International Proceedings in Informatics, LIPIcs; Том 64).Результаты исследований: Публикации в книгах, отчётах, сборниках, трудах конференций › статья в сборнике материалов конференции › Рецензирование
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TY - GEN
T1 - Complexity of distributions and average-case hardness
AU - Itsykson, Dmitry
AU - Knop, Alexander
AU - Sokolov, Dmitry
PY - 2016/12/1
Y1 - 2016/12/1
N2 - We address the following question in the average-case complexity: does there exists a language L such that for all easy distributions D the distributional problem (L, D) is easy on the average while there exists some more hard distribution D′ such that (L, D′) is hard on the average? We consider two complexity measures of distributions: the complexity of sampling and the complexity of computing the distribution function. For the complexity of sampling of distribution, we establish a connection between the above question and the hierarchy theorem for sampling distribution recently studied by Thomas Watson. Using this connection we prove that for every 0 < a < b there exist a language L, an ensemble of distributions D samplable in nlogb n steps and a linear-time algorithm A such that for every ensemble of distribution F that samplable in nloga n steps, A correctly decides L on all inputs from {0, 1}n except for a set that has infinitely small F-measure, and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}n for which B correctly decides L has infinitely small D-measure. In case of complexity of computing the distribution function we prove the following tight result: for every a > 0 there exist a language L, an ensemble of polynomial-time computable distributions D, and a linear-time algorithm A such that for every computable in na steps ensemble of distributions F, A correctly decides L on all inputs from {0, 1}n except for a set that has F-measure at most 2-n/2, and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}n for which B correctly decides L has D-measure at most 2-n+1.
AB - We address the following question in the average-case complexity: does there exists a language L such that for all easy distributions D the distributional problem (L, D) is easy on the average while there exists some more hard distribution D′ such that (L, D′) is hard on the average? We consider two complexity measures of distributions: the complexity of sampling and the complexity of computing the distribution function. For the complexity of sampling of distribution, we establish a connection between the above question and the hierarchy theorem for sampling distribution recently studied by Thomas Watson. Using this connection we prove that for every 0 < a < b there exist a language L, an ensemble of distributions D samplable in nlogb n steps and a linear-time algorithm A such that for every ensemble of distribution F that samplable in nloga n steps, A correctly decides L on all inputs from {0, 1}n except for a set that has infinitely small F-measure, and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}n for which B correctly decides L has infinitely small D-measure. In case of complexity of computing the distribution function we prove the following tight result: for every a > 0 there exist a language L, an ensemble of polynomial-time computable distributions D, and a linear-time algorithm A such that for every computable in na steps ensemble of distributions F, A correctly decides L on all inputs from {0, 1}n except for a set that has F-measure at most 2-n/2, and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}n for which B correctly decides L has D-measure at most 2-n+1.
KW - Average-case complexity
KW - Diagonalization
KW - Hierarchy theorem
KW - Sampling distributions
UR - http://www.scopus.com/inward/record.url?scp=85010790010&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ISAAC.2016.38
DO - 10.4230/LIPIcs.ISAAC.2016.38
M3 - Conference contribution
AN - SCOPUS:85010790010
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 38.1-38.12
BT - 27th International Symposium on Algorithms and Computation, ISAAC 2016
A2 - Hong, Seok-Hee
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 27th International Symposium on Algorithms and Computation, ISAAC 2016
Y2 - 12 December 2016 through 14 December 2016
ER -
ID: 49785539