Time hierarchies for heuristic algorithms
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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

Time Hierarchies for Heuristic Algorithms

Konstantin Pervyshev

UCSD


Outline

Outline

  • Introduction

    • known/unknown about time hierarchies

      & why heuristics

  • One sketch

    • time hierarchy for heuristics NP

      via error-correction


Introduction

Introduction


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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