Complexity of distributions and average-case hardness Dmitry - PowerPoint PPT Presentation

Complexity of distributions and average-case hardness Dmitry Sokolov joint work with Dmitry Itsykson and Alexander Knop St. Petersburg Department of V. A. Steklov Institute of Mathematics Problems in Theoretical Computer Science Moscow December 18-20, 2015 Sokolov D. | Complexity of distributions and average-case hardness 1/11

Definitions Ensembles of distributions Ensemble of distributions D = { D n } ∞ n = 1 . D ∈ Samp [ n k ] ⇔ there is a randomized O ( n k ) -time algorithm A such that D n and A ( 1 n ) are equally distributed. Samp [ n k ] . PSamp = � k Heuristic computations L is a language, D is an ensemble of distributions. ( L , D ) ∈ Heur δ DTime [ n k ] ⇔ there is O ( n k ) -time algorithm A such that x ← D n [ A ( x ) = L ( x )] > 1 − δ . Pr Heur δ DTime [ n k ] . Heur δ P = � k Sokolov D. | Complexity of distributions and average-case hardness 2/11

Definitions Ensembles of distributions Ensemble of distributions D = { D n } ∞ n = 1 . D ∈ Samp [ n k ] ⇔ there is a randomized O ( n k ) -time algorithm A such that D n and A ( 1 n ) are equally distributed. Samp [ n k ] . PSamp = � k Heuristic computations L is a language, D is an ensemble of distributions. ( L , D ) ∈ Heur δ DTime [ n k ] ⇔ there is O ( n k ) -time algorithm A such that x ← D n [ A ( x ) = L ( x )] > 1 − δ . Pr Heur δ DTime [ n k ] . Heur δ P = � k Sokolov D. | Complexity of distributions and average-case hardness 2/11

“Easy” problems Theorem (Gurevich and Shelah, 1987) Let HP denote the language of Hamiltonian graphs. Then ( HP , U ) ∈ Heur 2 O ( √ n ) DTime [ n ] . 1 Theorem (Babai, Erdos and Selkow, 1980) Let GI denote the language of pairs of isomorphic graphs. Then ( GI , U ) ∈ Heur √ n DTime [ n ] . 1 7 Sokolov D. | Complexity of distributions and average-case hardness 3/11

“Easy” problems Theorem (Gurevich and Shelah, 1987) Let HP denote the language of Hamiltonian graphs. Then ( HP , U ) ∈ Heur 2 O ( √ n ) DTime [ n ] . 1 Theorem (Babai, Erdos and Selkow, 1980) Let GI denote the language of pairs of isomorphic graphs. Then ( GI , U ) ∈ Heur √ n DTime [ n ] . 1 7 Sokolov D. | Complexity of distributions and average-case hardness 3/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Goal and Result Goal Result For every k there is a language L , For every 0 < a < b there is a ensemble D and small δ such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ PSamp ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] 3 for every R ∈ Samp [ n k ] we have that we have that ( L , R ) ∈ Heur δ DTime [ n ] . ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 4/11

Past and Present Gutfreund et al., 2007 Result If NP � BPP , there exists For every 0 < a < b there is a ensemble D such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ Samp [ quasi - poly ] ; 2 for every L ∈ NP -complete 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] and every α ( n ) = o ( 1 ) ( L , D ) / ∈ Heur α ( n ) BPP . we have that ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 5/11

Past and Present Gutfreund et al., 2007 Result If NP � BPP , there exists For every 0 < a < b there is a ensemble D such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ Samp [ quasi - poly ] ; 2 for every L ∈ NP -complete 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] and every α ( n ) = o ( 1 ) ( L , D ) / ∈ Heur α ( n ) BPP . we have that ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 5/11

Past and Present Gutfreund et al., 2007 Result If NP � BPP , there exists For every 0 < a < b there is a ensemble D such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ Samp [ quasi - poly ] ; 2 for every L ∈ NP -complete 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] and every α ( n ) = o ( 1 ) ( L , D ) / ∈ Heur α ( n ) BPP . we have that ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 5/11

Past and Present Gutfreund et al., 2007 Result If NP � BPP , there exists For every 0 < a < b there is a ensemble D such that: language L , ensemble D and δ = o ( 1 ) such that: 1 D ∈ Samp [ n log b n ] ; 1 D ∈ Samp [ quasi - poly ] ; 2 for every L ∈ NP -complete 2 ( L , D ) / ∈ Heur 1 − δ P ; 3 for every R ∈ Samp [ n log a n ] and every α ( n ) = o ( 1 ) ( L , D ) / ∈ Heur α ( n ) BPP . we have that ( L , R ) ∈ Heur δ DTime [ n ] . Sokolov D. | Complexity of distributions and average-case hardness 5/11