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Time Hierarchies Time Hierarchies for for Heuristic Algorithms Heuristic Algorithms Konstantin Pervyshev Konstantin Pervyshev UCSD UCSD
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Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Dec 20, 2015

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Page 1: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Time HierarchiesTime Hierarchies for for

Heuristic Algorithms Heuristic Algorithms

Konstantin PervyshevKonstantin Pervyshev

UCSDUCSD

Page 2: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

OutlineOutline

• IntroductionIntroduction– known/unknown about time hierarchiesknown/unknown about time hierarchies

& why heuristics& why heuristics

• One sketchOne sketch– time hierarchy for heuristics NPtime hierarchy for heuristics NP

via error-correctionvia error-correction

Page 3: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

IntroductionIntroduction

Page 4: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Time HierarchiesTime Hierarchies

• Problems having odd complexityProblems having odd complexity– O(nO(n100100) and not much less) and not much less

• Proven forProven for– any syntactic model (like P & NP)any syntactic model (like P & NP)– no semantic model (like BPP)no semantic model (like BPP)

Page 5: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Syntactic vs. SemanticSyntactic vs. Semantic

• Syntactic modelsSyntactic models– Syntactically correct machinesSyntactically correct machines– Examples: P, NP, coNP, ParityPExamples: P, NP, coNP, ParityP

• Semantic modelsSemantic models– Additional semantic constraintsAdditional semantic constraints– Examples: BPP, AM, UPExamples: BPP, AM, UP

Page 6: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Open QuestionOpen Question

• Time hierarchies for semantic modelsTime hierarchies for semantic models– probabilistic algorithms (BPP / RP / ZPP)probabilistic algorithms (BPP / RP / ZPP)– Arthur-Merlin & Merlin-Arthur games (AM / Arthur-Merlin & Merlin-Arthur games (AM /

MA)MA)– unambigous machines (UP)unambigous machines (UP)– other semantic classesother semantic classes

Page 7: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Non-Traditional SettingsNon-Traditional Settings

Slightly non-uniformSlightly non-uniformalgorithmsalgorithms[Barak’02][Barak’02]

HeuristicHeuristicalgorithmsalgorithms

[Fortnow,Santhanam’04][Fortnow,Santhanam’04]

Time Hierarchies in Other Settings

input x of length n input x of length n + (short) advice a+ (short) advice ann

make mistakes on make mistakes on δ(n)-fraction of inputsδ(n)-fraction of inputs

Page 8: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Time Hierarchies forTime Hierarchies for1-Bit Non-Uniform Algorithms1-Bit Non-Uniform Algorithms

• Syntactic modelsSyntactic models– any model/1any model/1

• Semantic modelsSemantic models– BPP/1BPP/1 & & BQP/1BQP/1 [Fortnow, [Fortnow,

Santhanam’04]Santhanam’04]– RP/1RP/1 [Fortnow, Santhanam, [Fortnow, Santhanam,

Trevisan’05]Trevisan’05]– any model/1any model/1 [van Melkebeek, P. ’06] [van Melkebeek, P. ’06]

Page 9: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Time Hierarchies forTime Hierarchies forHeuristic AlgorithmsHeuristic Algorithms

• Syntactic modelsSyntactic models– any model closed under complementany model closed under complement– UnknownUnknown: those that are not closed: those that are not closed

(think of (think of heurNPheurNP))

• Semantic modelsSemantic models– heurBPPheurBPP & & heurBQPheurBQP

[Fortnow, Santhanam’04][Fortnow, Santhanam’04]– UnknownUnknown: any other: any other

Page 10: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Scope of This TalkScope of This Talk

Slightly non-uniformSlightly non-uniformDONEDONE

HeuristicHeuristicTHIS WORKTHIS WORK

Time Hierarchies in Other Settings

Page 11: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Our Results:Our Results:More Time Hierarchies for More Time Hierarchies for HeuristicsHeuristics• Syntactic models:Syntactic models:

– any model closed under majorityany model closed under majority

(NP, co-NP, alternation classes)(NP, co-NP, alternation classes)

• Semantic models:Semantic models:– some more probabilistic modelssome more probabilistic models

(AM, MA, a stronger hierarchy for BPP)(AM, MA, a stronger hierarchy for BPP)

Page 12: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Our ApproachOur Approach

(on the example of heuristic NP)(on the example of heuristic NP)

Page 13: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Hierarchies for NPHierarchies for NP

NPNP not subset of not subset of NTime[n]NTime[n]– poly-time poly-time NN vs. linear-time vs. linear-time MMii

– for some for some xx, , N(x) ≠ MN(x) ≠ Mii(x)(x)

NPNP not subset of not subset of heurheur1/2+1/n1/2+1/na a NTime[n]NTime[n]

– whatever whatever MMii, for some , for some nn, ,

PrPrx in {0,1}x in {0,1}nn [N(x) ≠ M [N(x) ≠ Mii(x)] > 1/2-1/n(x)] > 1/2-1/naa

Page 14: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Non-Heuristic Case:Non-Heuristic Case:ReviewReview

• Assume that for every Assume that for every xx, , N(x) = N(x) = MMii(x) (x)

• Construct Construct NN so that for some so that for some xx,,

N(x) ≠ MN(x) ≠ Mii(x) (x)

• Hence, a contradictionHence, a contradiction

Page 15: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Non-Heuristic Case:Non-Heuristic Case:ReviewReview

we wantwe wantN(xN(xnn) = b) = b

xxkk = “0…0” = “0…0” of length k of length k

xn xn+1 x2nx2n - 1xn+2 x2n - 2. . . .

b = ¬ Mb = ¬ Mii(x(xnn))

we canwe canN(xN(x22nn) = b) = b

Page 16: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Non-Heuristic Case:Non-Heuristic Case:ReviewReview

we needwe needN(xN(xkk) = N(x) = N(xk+1k+1))

MMii(x(xk+1k+1) = ) = N(xN(xk+1k+1))

(by assumption)(by assumption)

N(xN(xkk) = M) = Mii(x(xk+1k+1))(by construction)(by construction)

xn xn+1 x2nx2n - 1xn+2 x2n - 2. . . .

Page 17: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Heuristic CaseHeuristic Case

weaker assumptionweaker assumption

for any n, for any n,

PrPrx in {0,1}x in {0,1}nn [M [Mii(x) = N(x)] > (x) = N(x)] > 1/2+1/n1/2+1/naa

Page 18: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Transmission FailureTransmission Failure

we needwe needN(xN(xkk) = N(x) = N(xk+1k+1))

MMii(x(xk+1k+1) ) ?? N(x N(xk+1k+1))(by assumption)(by assumption)

N(xN(xkk) = M) = Mii(x(xk+1k+1))(by construction)(by construction)

xn xn+1 x2nx2n - 1xn+2 x2n - 2. . . .

Page 19: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Repairing the ChannelRepairing the Channel

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

Answer: Answer: yesyes, ,

use error-correction!use error-correction!

• Repetition code ( Repetition code ( b b … b b b b … b b ))

Page 20: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

High-Level ViewHigh-Level View

we wantwe wantN(x) = bN(x) = bfor anyfor anyx x inin Y Ynn

YYkk = {0,1} = {0,1}kk

Yn Yn+1 Y2nY2n - 1Yn+2 Y2n - 2. . . .

b = ¬ maj b = ¬ maj x in Yx in Ynn {M{Mii(x)}(x)}

we canwe canN(x) = bN(x) = bfor anyfor anyx x inin Y Y22nn

Page 21: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

One Step of TransmissionOne Step of Transmission

N(x) = bN(x) = bfor any for any xx in in YYk+1k+1

““codeword of codeword of bb””

maj maj x in Yx in Yk+1k+1 {M {Mii(x)} = b(x)} = b

““corrupted message”corrupted message”

N(x) = b N(x) = b for any for any xx in in YYkk

““recovered codeword of recovered codeword of bb””

Page 22: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Codeword RecoveryCodeword Recovery

maj maj x in Yx in Yk+1k+1 {M {Mii(x)} = b(x)} = b

““corrupted message”corrupted message”

N(x) = b N(x) = b (almost)(almost) for any for any xx in in YYkk

““recovered codeword of recovered codeword of bb””

Expanders

Q.E.D.

Page 23: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

A few words about A few words about heuristic BPPheuristic BPP

heurheur1-1/n1-1/naaBPPBPP

not subset ofnot subset of

heurheur1/2+1/n1/2+1/na a BPTime[n]BPTime[n]

Page 24: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Heuristic Heuristic BPPBPP

• More easy:More easy:

compute majority by estimatingcompute majority by estimating

θ ≈ θ ≈ PrPrx in Yx in Yk+1k+1 [M [Mii(x) = 1](x) = 1]

& comparing θ to a threshold ½& comparing θ to a threshold ½

• More difficult:More difficult:

NN should be semantically correct; should be semantically correct;

on different inputs, use different thresholdson different inputs, use different thresholds

Page 25: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

ResultsResults

• NPNP

not subset ofnot subset of

heurheur1/2+1/n1/2+1/na a NTime[n]NTime[n]

• heurheur1-1/n1-1/na a AM/MA/BPPAM/MA/BPP

not subset ofnot subset of

heurheur1/2+1/n1/2+1/na a AM/MA/BPTime[n]AM/MA/BPTime[n]

Page 26: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Open QuestionsOpen Questions

• Time hierarchies for heuristic Time hierarchies for heuristic RP/ZPPRP/ZPP

• heurheur1-ε 1-ε NP NP vs. vs. heurheur½½ NTime[n] NTime[n] &&

heurheur1-ε 1-ε BPP BPP vs. vs. heurheur½½ BPTime[n]BPTime[n]

• Time hierarchies for non-heuristic Time hierarchies for non-heuristic semantic modelssemantic models

Page 27: Time Hierarchies for Heuristic Algorithms Konstantin Pervyshev UCSD.

Have a safe trip!Have a safe trip!

pervyshev @ cs.ucsd.edupervyshev @ cs.ucsd.edu