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Time Hierarchies for Heuristic Algorithms. Konstantin Pervyshev UCSD. Outline. Introduction known/unknown about time hierarchies & why heuristics One sketch time hierarchy for heuristics NP via error-correction. Introduction. Time Hierarchies. Problems having odd complexity

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## Time Hierarchies for Heuristic Algorithms

Konstantin Pervyshev

UCSD

### Outline

• Introduction

• known/unknown about time hierarchies

& why heuristics

• One sketch

• time hierarchy for heuristics NP

via error-correction

## Introduction

### Time Hierarchies

• Problems having odd complexity

• O(n100) and not much less

• Proven for

• any syntactic model (like P & NP)

• no semantic model (like BPP)

### Syntactic vs. Semantic

• Syntactic models

• Syntactically correct machines

• Examples: P, NP, coNP, ParityP

• Semantic models

• Additional semantic constraints

• Examples: BPP, AM, UP

### Open Question

• Time hierarchies for semantic models

• probabilistic algorithms (BPP / RP / ZPP)

• Arthur-Merlin & Merlin-Arthur games (AM / MA)

• unambigous machines (UP)

• other semantic classes

### Non-Traditional Settings

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

### Time Hierarchies for1-Bit Non-Uniform Algorithms

• 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]

### Time Hierarchies forHeuristic Algorithms

• 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

### Scope of This Talk

Time Hierarchies in Other Settings

Slightly non-uniform

DONE

Heuristic

THIS WORK

### Our Results:More Time Hierarchies for Heuristics

• Syntactic models:

• any model closed under majority

(NP, co-NP, alternation classes)

• Semantic models:

• some more probabilistic models

(AM, MA, a stronger hierarchy for BPP)

## Our Approach

(on the example of heuristic NP)

### Hierarchies for 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

### Non-Heuristic Case:Review

• 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

### Non-Heuristic Case:Review

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

### Non-Heuristic Case:Review

we need

N(xk) = N(xk+1)

N(xk) = Mi(xk+1)

(by construction)

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

(by assumption)

### Heuristic Case

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

### Transmission Failure

we need

N(xk) = N(xk+1)

N(xk) = Mi(xk+1)

(by construction)

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

(by assumption)

### Repairing the Channel

• 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

### High-Level View

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

### One Step of Transmission

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”

### Codeword Recovery

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]

### Heuristic BPP

• 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

### Results

• 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]

### Open Questions

• 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