Complexity of distributions and average-case hardness Authors: - PowerPoint PPT Presentation

ISAAC 2016 Complexity of distributions and average-case hardness Authors: Dmitry Itsykson, Alexander Knop, Dmitry Sokolov Institute: St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences

What computer scientists want SAT IS EASY (P = NP) There is a plynomial-time algorithm such that for all Boolean formulas it decides correctly if they are satisfjable or not. SAT IS HARD (P NP) For any plynomial-time algorithm, there is a Boolean formula such that this algorithm decides if it is satisfjable or not incorrectly. INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 2

What computer scientists want SAT IS EASY (P = NP) There is a plynomial-time algorithm such that for all Boolean formulas it decides correctly if they are satisfjable or not. For any plynomial-time algorithm, there is a Boolean formula such that this algorithm decides if it is satisfjable or not incorrectly. INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 2 SAT IS HARD (P ̸ = NP)

What computer scientists really want SAT IS EASY There is a plynomial-time algorithm such that for almost all Boolean formulas it decides correctly if they are satisfjable or not. SAT IS HARD For any plynomial-time algorithm there are many Boolean formulas such that this algorithm decides if they are satisfjable or not incorrectly. INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 3

What computer scientists really want SAT IS EASY There is a plynomial-time algorithm such that for almost all Boolean formulas it decides correctly if they are satisfjable or not. SAT IS HARD For any plynomial-time algorithm there are many Boolean formulas such that this algorithm decides if they are satisfjable or not incorrectly. INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 3

What distribution? SAT IS EASY There is a plynomial-time algorithm such that for almost all Boolean formulas it decides correctly if they are satisfjable or not. SAT IS HARD For any plynomial-time algorithm there are many formulas such that this algorithm decides if they are satisfjable or not incorrectly. LI AND VITANYI, 1992 There is a distribution on strings such that for any language L , if n fraction of L is decidable in polynomial time, then L is decidable in polynomial time. INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 4

What distribution? SAT IS EASY There is a plynomial-time algorithm such that for almost all Boolean formulas it decides correctly if they are satisfjable or not. SAT IS HARD For any plynomial-time algorithm there are many formulas such that this algorithm decides if they are satisfjable or not incorrectly. LI AND VITANYI, 1992 of L is decidable in polynomial time, then L is decidable in polynomial time. INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 4 There is a distribution on strings such that for any language L , if 1 − 1 n 3 fraction

n x is a cumulative Natural classes of distributions SAMPLABLE DISTRIBUTIONS COMPUTABLE DISTRIBUTIONS An ensamble of distributions D is samplable in time f n ifg there is a f n -time algorithm such that for any n , function x A distribution function of D n . INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 5 An ensamble of distributions D is samplable in time f ( n ) ifg there is a f ( n ) -time algorithm such that for any n distributions D n and A (1 n ) are equally distributed.

Natural classes of distributions SAMPLABLE DISTRIBUTIONS COMPUTABLE DISTRIBUTIONS distribution function of D n . INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 5 An ensamble of distributions D is samplable in time f ( n ) ifg there is a f ( n ) -time algorithm such that for any n distributions D n and A (1 n ) are equally distributed. An ensamble of distributions D is samplable in time f ( n ) ifg there is a f ( n ) -time algorithm such that for any n , function x → A (1 n , x ) is a cumulative

Classes of computations HEURISTICALLY DECIDABLE IN POLYNOMIAL TIME Let L is a language and D is an ensemble of distributions. We call INTRODUCTION | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 6 distributional problem ( L , D ) heuristically decidable in polynomial time with error ϵ ( n ) ( ( L , D ) ∈ Heur ϵ ( n ) P ) ifg there is a polynomial time algorithm A such that Pr x ← D n [ A ( x ) ̸ = L ( x )] ≤ ϵ ( n ) .

Heur BPP and DSamp n k holds L R Heur BPTime n k . PSamp and a language L such that Heur P and for any R DSamp n k holds L R Heur P . STATEMENT | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov L D Is there a distribution D Complexity of distribution DUAL QUESTION? FOLKLORE for any R there is a language L such that L U and For any k ITSYKSON, K, AND SOKOLOV, 2015 7 For any k > 0 and δ there is a language L such that ( L , U ) ∈ Heur δ P and for any R holds ( L , R ) ̸∈ Heur 1 − δ DTime ( n k ) .

PSamp and a language L such that Heur P and for any R DSamp n k holds L R Heur P . Complexity of distribution STATEMENT | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov L D Is there a distribution D FOLKLORE DUAL QUESTION? ITSYKSON, K, AND SOKOLOV, 2015 7 For any k > 0 and δ there is a language L such that ( L , U ) ∈ Heur δ P and for any R holds ( L , R ) ̸∈ Heur 1 − δ DTime ( n k ) . For any k > 0 and δ there is a language L such that ( L , U ) ∈ Heur δ BPP and for any R ∈ DSamp ( n k ) holds ( L , R ) ̸∈ Heur 1 2 − δ BPTime ( n k ) .

Complexity of distribution FOLKLORE ITSYKSON, K, AND SOKOLOV, 2015 DUAL QUESTION? STATEMENT | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 7 For any k > 0 and δ there is a language L such that ( L , U ) ∈ Heur δ P and for any R holds ( L , R ) ̸∈ Heur 1 − δ DTime ( n k ) . For any k > 0 and δ there is a language L such that ( L , U ) ∈ Heur δ BPP and for any R ∈ DSamp ( n k ) holds ( L , R ) ̸∈ Heur 1 2 − δ BPTime ( n k ) . Is there a distribution D ∈ PSamp and a language L such that ( L , D ) ̸∈ Heur 1 − δ P and for any R ∈ DSamp ( n k ) holds ( L , R ) ∈ Heur δ P .

Let HP denote the language of Hamiltonian graphs. Then HP U Heur n DTime n . Let GI denote the language of pairs of isomorphic graphs. Then GI U Heur n DTime n . Known results STATEMENT | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov O BABAI, ERDOS, AND SELKOW, 1980 DUAL QUESTION? GUREVICH AND SHELAH, 1987 8 Is there a distribution D ∈ PSamp and a language L such that ( L , D ) ̸∈ Heur 1 − δ P and for any R ∈ DSamp ( n k ) holds ( L , R ) ∈ Heur δ P .

Let GI denote the language of pairs of isomorphic graphs. Then GI U Heur n DTime n . Known results DUAL QUESTION? GUREVICH AND SHELAH, 1987 BABAI, ERDOS, AND SELKOW, 1980 STATEMENT | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 8 Is there a distribution D ∈ PSamp and a language L such that ( L , D ) ̸∈ Heur 1 − δ P and for any R ∈ DSamp ( n k ) holds ( L , R ) ∈ Heur δ P . Let HP denote the language of Hamiltonian graphs. Then ( HP , U ) ∈ Heur 2 O ( √ n ) DTime ( n ) . 1

Known results DUAL QUESTION? STATEMENT | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov BABAI, ERDOS, AND SELKOW, 1980 8 GUREVICH AND SHELAH, 1987 Is there a distribution D ∈ PSamp and a language L such that ( L , D ) ̸∈ Heur 1 − δ P and for any R ∈ DSamp ( n k ) holds ( L , R ) ∈ Heur δ P . Let HP denote the language of Hamiltonian graphs. Then ( HP , U ) ∈ Heur 2 O ( √ n ) DTime ( n ) . 1 Let GI denote the language of pairs of isomorphic graphs. Then ( GI , U ) ∈ Heur √ n DTime ( n ) . 1 7

PSamp , and a language Heur P and L R Heur P for any DSamp n k . PSamp such that for DSamp n k the statistical distance between R and D is at least RESULTS | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov n . any R , a distribution D n There is a function Equal quesition R STATISTICAL DISTANCE L such that L D , a distribution D n There is a function For any k the following two statements are equal: EQUIVALENT RESTATEMENT 9 A statistical distance between D n and R n is ∆( D n , R n ) = max S ⊆{ 0 , 1 } n | D n ( S ) − R n ( S ) | .

Equal quesition STATISTICAL DISTANCE EQUIVALENT RESTATEMENT For any k the following two statements are equal: RESULTS | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov 9 A statistical distance between D n and R n is ∆( D n , R n ) = max S ⊆{ 0 , 1 } n | D n ( S ) − R n ( S ) | . ▶ There is a function δ ( n ) → 0 , a distribution D ∈ PSamp , and a language L such that ( L , D ) ̸∈ Heur 1 − δ P and ( L , R ) ∈ Heur δ P for any R ∈ DSamp ( n k ) . ▶ There is a function δ ( n ) → 0 , a distribution D ∈ PSamp such that for any R ∈ DSamp ( n k ) the statistical distance between R and D is at least 1 − δ ( n ) .

DSamp n log c n PSamp where Hierarchies for distributions that RESULTS | Dmitry Itsykson, Alexander Knop, Dmitry Sokolov . n n for any R D R such WATSON, 2013 there is a distribution D For any constant c and ITSYKSON, K, SOKOLOV, 2016 10 For any constant k and ϵ > 0 there is a distribution D ∈ PSamp such that k + ϵ for any R ∈ DSamp ( n k ) . ∆( D , R ) ≥ 1 − 1