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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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Presentation Transcript
outline
Outline
  • Introduction
    • known/unknown about time hierarchies

& why heuristics

  • One sketch
    • time hierarchy for heuristics NP

via error-correction

time hierarchies
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 vs. Semantic
  • Syntactic models
    • Syntactically correct machines
    • Examples: P, NP, coNP, ParityP
  • Semantic models
    • Additional semantic constraints
    • Examples: BPP, AM, UP
open question
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
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 for 1 bit non uniform algorithms
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 for heuristic algorithms1
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
Scope of This Talk

Time Hierarchies in Other Settings

Slightly non-uniform

DONE

Heuristic

THIS WORK

our results more time hierarchies for heuristics
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

Our Approach

(on the example of heuristic NP)

hierarchies for 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
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
non heuristic case review1

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

non heuristic case review2

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
Heuristic Case

weaker assumption

for any n,

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

transmission failure

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
Repairing the Channel
  • Question: can we repair the channel ?

Answer: yes,

use error-correction!

  • Repetition code ( b b … b b )
high level view

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
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
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

A few words about heuristic BPP

heur1-1/naBPP

not subset of

heur1/2+1/na BPTime[n]

heuristic bpp
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
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
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

Have a safe trip!

pervyshev @ cs.ucsd.edu

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