Time Hierarchies for Heuristic Algorithms

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Time Hierarchies for Heuristic Algorithms

• Konstantin Pervyshev
• UCSD

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Outline

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

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Introduction
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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)

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Syntactic vs. Semantic

• Syntactic models
• Syntactically correct machines
• Examples P, NP, coNP, ParityP
• Semantic models
• Additional semantic constraints
• Examples BPP, AM, UP

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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

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Non-Traditional Settings
Time Hierarchies in Other Settings
Slightly non-uniform algorithms Barak02
Heuristic algorithms Fortnow,Santhanam04
input x of length n (short) advice an
make mistakes on d(n)-fraction of inputs
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Time Hierarchies for1-Bit Non-Uniform Algorithms

• Syntactic models
• any model/1
• Semantic models
• BPP/1 BQP/1 Fortnow, Santhanam04
• RP/1 Fortnow, Santhanam, Trevisan05
• any model/1 van Melkebeek, P. 06

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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, Santhanam04
• Unknown any other

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Scope of This Talk
Time Hierarchies in Other Settings
Slightly non-uniform DONE
Heuristic THIS WORK
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Our ResultsMore 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)

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Our Approach

• (on the example of heuristic NP)

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Hierarchies for NP

• NP not subset of NTimen
• poly-time N vs. linear-time Mi
• for some x, N(x) ? Mi(x)
• NP not subset of heur1/21/na NTimen
• whatever Mi, for some n,
• Prx in 0,1n N(x) ? Mi(x) gt 1/2-1/na

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Non-Heuristic CaseReview

• Assume that for every x, N(x) Mi(x)
• Construct N so that for some x,
• N(x) ? Mi(x)
• Hence, a contradiction

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Non-Heuristic CaseReview
xk 00 of length k
b Mi(xn)
we want N(xn) b
we can N(x2n) b
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Non-Heuristic CaseReview
we need N(xk) N(xk1)
Mi(xk1) N(xk1) (by assumption)
N(xk) Mi(xk1) (by construction)
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Heuristic Case

• weaker assumption
• for any n,
• Prx in 0,1n Mi(x) N(x) gt 1/21/na

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Transmission Failure
we need N(xk) N(xk1)
Mi(xk1) ? N(xk1) (by assumption)
N(xk) Mi(xk1) (by construction)
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Repairing the Channel

• Question can we repair the channel ?
• Answer yes,
• use error-correction!
• Repetition code ( b b b b )

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High-Level View
Yk 0,1k
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
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One Step of Transmission
N(x) b for any x in Yk1 codeword of b
N(x) b for any x in Yk recovered codeword of
b
maj x in Yk1 Mi(x) b corrupted message
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Codeword Recovery
N(x) b (almost) for any x in Yk recovered
codeword of b
Expanders
maj x in Yk1 Mi(x) b corrupted message
Q.E.D.
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A few words about heuristic BPP

• heur1-1/naBPP
• not subset of
• heur1/21/na BPTimen

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Heuristic BPP

• More easy
• compute majority by estimating
• ? Prx in Yk1 Mi(x) 1
• comparing ? to a threshold ½
• More difficult
• N should be semantically correct
• on different inputs, use different thresholds

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Results

• NP
• not subset of
• heur1/21/na NTimen
• heur1-1/na AM/MA/BPP
• not subset of
• heur1/21/na AM/MA/BPTimen

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Open Questions

• Time hierarchies for heuristic RP/ZPP
• heur1-e NP vs. heur½ NTimen
• heur1-e BPP vs. heur½ BPTimen
• Time hierarchies for non-heuristic semantic
  models

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Have a safe trip!

• pervyshev _at_ cs.ucsd.edu