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Time Hierarchies for Heuristic Algorithms

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Time Hierarchies for Heuristic Algorithms

Konstantin Pervyshev

UCSD

- Introduction
- known/unknown about time hierarchies
& why heuristics

- known/unknown about time hierarchies
- One sketch
- time hierarchy for heuristics NP
via error-correction

- time hierarchy for heuristics NP

Introduction

- Problems having odd complexity
- O(n100) and not much less

- Proven for
- any syntactic model (like P & NP)
- no semantic model (like BPP)

- Syntactic models
- Syntactically correct machines
- Examples: P, NP, coNP, ParityP

- Semantic models
- Additional semantic constraints
- Examples: BPP, AM, UP

- Time hierarchies for semantic models
- probabilistic algorithms (BPP / RP / ZPP)
- Arthur-Merlin & Merlin-Arthur games (AM / MA)
- unambigous machines (UP)
- other semantic classes

Time Hierarchies in Other Settings

Slightly non-uniform

algorithms

[Barak’02]

Heuristic

algorithms

[Fortnow,Santhanam’04]

input x of length n

+ (short) advice an

make mistakes on

δ(n)-fraction of inputs

- Syntactic models
- any model/1

- Semantic models
- BPP/1 & BQP/1 [Fortnow, Santhanam’04]
- RP/1 [Fortnow, Santhanam, Trevisan’05]
- any model/1 [van Melkebeek, P. ’06]

- Syntactic models
- any model closed under complement
- Unknown: those that are not closed
(think of heurNP)

- Semantic models
- heurBPP & heurBQP
[Fortnow, Santhanam’04]

- Unknown: any other

- heurBPP & heurBQP

Time Hierarchies in Other Settings

Slightly non-uniform

DONE

Heuristic

THIS WORK

- Syntactic models:
- any model closed under majority
(NP, co-NP, alternation classes)

- any model closed under majority
- Semantic models:
- some more probabilistic models
(AM, MA, a stronger hierarchy for BPP)

- some more probabilistic models

Our Approach

(on the example of heuristic NP)

NP not subset of NTime[n]

- poly-time N vs. linear-time Mi
- for some x, N(x) ≠ Mi(x)
NP not subset of heur1/2+1/na NTime[n]

- whatever Mi, for some n,
Prx in {0,1}n [N(x) ≠ Mi(x)] > 1/2-1/na

- Assume that for every x, N(x) = Mi(x)
- Construct N so that for some x,
N(x) ≠ Mi(x)

- Hence, a contradiction

xn

xn+1

xn+2

. . . .

x2n - 2

x2n - 1

x2n

xk = “0…0” of length k

b = ¬ Mi(xn)

we want

N(xn) = b

we can

N(x2n) = b

xn

xn+1

xn+2

. . . .

x2n - 2

x2n - 1

x2n

we need

N(xk) = N(xk+1)

N(xk) = Mi(xk+1)

(by construction)

Mi(xk+1) = N(xk+1)

(by assumption)

weaker assumption

for any n,

Prx in {0,1}n [Mi(x) = N(x)] > 1/2+1/na

xn

xn+1

xn+2

. . . .

x2n - 2

x2n - 1

x2n

we need

N(xk) = N(xk+1)

N(xk) = Mi(xk+1)

(by construction)

Mi(xk+1) ? N(xk+1)

(by assumption)

- Question: can we repair the channel ?
Answer: yes,

use error-correction!

- Repetition code ( b b … b b )

Yn

Yn+1

Yn+2

. . . .

Y2n - 2

Y2n - 1

Y2n

Yk = {0,1}k

b = ¬ maj x in Yn{Mi(x)}

we want

N(x) = b

for any

x in Yn

we can

N(x) = b

for any

x in Y2n

N(x) = b

for any x in Yk

“recovered codeword of b”

N(x) = b

for any x in Yk+1

“codeword of b”

maj x in Yk+1 {Mi(x)} = b

“corrupted message”

N(x) = b

(almost) for any x in Yk

“recovered codeword of b”

Expanders

maj x in Yk+1 {Mi(x)} = b

“corrupted message”

Q.E.D.

A few words about heuristic BPP

heur1-1/naBPP

not subset of

heur1/2+1/na BPTime[n]

- More easy:
compute majority by estimating

θ ≈ Prx in Yk+1 [Mi(x) = 1]

& comparing θ to a threshold ½

- More difficult:
N should be semantically correct;

on different inputs, use different thresholds

- NP
not subset of

heur1/2+1/na NTime[n]

- heur1-1/na AM/MA/BPP
not subset of

heur1/2+1/na AM/MA/BPTime[n]

- Time hierarchies for heuristic RP/ZPP
- heur1-ε NP vs. heur½NTime[n] &
heur1-ε BPP vs. heur½BPTime[n]

- Time hierarchies for non-heuristic semantic models

Have a safe trip!

pervyshev @ cs.ucsd.edu