Time Hierarchies for Heuristic Algorithms

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

Time Hierarchies for Heuristic Algorithms