the art of reduction (or, depth through breadth)
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The art of reduction (or, Depth through Breadth). Avi Wigderson Institute for Advanced Study Princeton. How difficult are the problems we want to solve? Problem A ? Problem B ? Reduction: An efficient algorithm for A implies an efficient algorithm for B. - PowerPoint PPT PresentationTRANSCRIPT
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The art of reduction
(or, Depth through Breadth)
Avi WigdersonInstitute for Advanced
Study Princeton
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How difficult are the problems we want to solve?
Problem A ? Problem B ?
Reduction: An efficient algorithm for A implies an efficient algorithm for B
If A is easy then B is easy
If B is hard then A is hard
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Constraint Satisfaction I
AI, Operating Systems, Compilers, Verification, Scheduling, …
3-SAT: Find a Boolean assignment to
the xi which satisfies all constraints
(x1x3 x7) (x1x2 x4) (x5x3 x6) …
3-COLORING: Find a vertex coloringusing which satisfies all constraints
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Classic reductions Thm[Cook, Levin ’71] 3-SAT is NP-complete
Thm[Karp 72]: 3-COLORING is NP-completeProof: If 3-COLORING easy then 3-SAT easy
formula graph
satisfying legalassignment coloring
Claim: In every legal 3-coloring, z = x y
F
y x
z OR-gadget
T
x y
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NP - completeness
Papadimutriou: “The largest intellectual export of CS to Math & Science.”
“2000 titles containing ‘NP-complete’in Physics and Chemistry alone”
“We use it to argue computational difficulty. They to argue structural nastiness”
Most NP-completeness reductions are “gadget” reductions.
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Constraint Satisfaction IIAI, Programming languages, Compilers,
Verification, Scheduling, … 3-SAT: Find a Boolean assignment to the
xi which satisfies most constraints
(x1x3 x7) (x1x2 x4) (x5x3 x6) …
Trivial to satisfy 7/8 of all constraints!(simply choose a random assignment)How much better can we do efficiently?
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Hardness of approximation
Thm[Hastad 01] Satisfying 7/8+ fraction of all constraints in 3-SAT is NP-hard
Proof: The PCP theorem [Arora-Safra, Arora-Lund-Motwani-Sudan-Szegedy]
& much more [Feige-Goldwasser-Lovasz-Safra-Szegedy, Hastad, Raz…]
Nontrivial approx exact solution of 3-SAT of 3-SAT
Interactive proofs
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3-SAT is 7/8+ hard
3-SAT is .99 hard
NP3-SAT is hard
Influencesin Booleanfunctions
DistributedComputing
2 DecadesDecade
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How to minimize players’ influence
Function InfluenceParity 1
Thm [Kahn—Kalai—Linial] : For every Boolean function
on n bits, some player has influence > (log n)/n(uses Functional & Harmonic Analysis)
P parity
Mmajority
M M M
M
Majority 1/7Iterated Majority 1/8
Public Information Model [BenOr-Linial] :
Joint random coin flipping / voting Every good player flips a coin, then combine using function f.Which f is best?
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3-SAT is 7/8+ hard
3-SAT is .99 hard
IP NP
3-SAT hard
Optimization Approx Algorithms
ArithmetizationProgramChecking
2IP
PCP
Cryptography Zero-knowledge
IP=PSPACE
2IP=NEXP
PCP =NP
ProbabilisticInteractiveProof systems
IP = Interactive Proofs2IP = 2-prover Interactive ProofsPCP = Probabilistically Checkable Proofs
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IP = (Probabilistic) Interactive Proofs[Babai, Goldwasser-Micali-Rackoff]
ProverAccept (x,w)iff xL
Accept (x,q,a)whp iff xL
Prover
Verifier
w
q1
a1……qr
ar
Wiles Referee
Socrates
Verifier (probabilistic)
Student
L NP
L IP
x
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ZK IP: Zero-Knowledge Interactive Proofs[Goldwasser-Micali-Rackoff]
Accept (x,q,a)whp iff xL
Prover q1
a1……qr
arSocrates
Verifier
Student
L ZK IP
x
Gets no otherInformation !
Thm [Goldreich-Micali-Wigderson] :If 1-way functions exist, NP ZK IP(Every proof can be made into a ZK proof
!)Thm [Ostrovsky-Wigderson] ZK 1WF
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2IP: 2-Prover Interactive Proofs[BenOr-Goldwasser-Kilian-Wigderson]
Verif. accept (x,q,a,p,b) whp iff xL
Prover1 q1
a1……qr
arSocrates
Verifier
Student
L2IP:
x
p1
b1……pr
br
Prover2
Plato
Theorem [BGKW] : NP ZK 2IP
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What is the power of Randomness and Interaction in Proofs?
NP
IP
2IPTrivial inclusions IP PSPACE 2IP NEXP
Few nontrivial examplesGraph non isomorphism
……Few years of stalemate
Polynomial SpaceNondeterministicExponential Time
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Meanwhile, in a different galaxy…Self testing & correcting programs[Blum-Kannan, Blum-Luby-Rubinfeld]
Given a program P for a function g : Fn Fand input x, compute g(x) correctlywhen P errs on (unknown) 1/8 of Fn
Easy case g is linear: g(w+z)=g(w)+g(z)Given x, Pick y at random, and compute P(x+y)-P(y) x Proby [P(x+y)-P(y) g(x)]
1/4Other functions?- The Permanent [BF,L]- Low degree polynomials
Fn
P errsx+y y
x
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Determinant & Permanent
X=(xij) is an nxn matrix over F
Two polynomials of degree n:
Determinant: d(X)= Sn sgn() i[n] Xi(i)
Permanent: g(X)= Sn i[n] Xi(i)
Thm[Gauss] Determinant is easyThm[Valiant] Permanent is hard
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The Permanent is self-correctable[Beaver-Feigenbaum, Lipton]
Given X, pick Y at random, and computeP(X+1Y),P(X+2Y),… P(X+(n+1)Y) WHP g(X+1Y),g(X+2Y),… g(X+(n+1)Y)
But g(X+uY)=h(u) is apolynomial of deg n.Interpolate to geth(0)=g(X)
g(X) = Sn i[n] Xi(i)
Fnxn
P errs
x+3yx+2y
xx+y
|| ||||
P errs on 1/(8n)
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Hardness amplificationX=(Xij) matrix, Per(X)= Sn i[n] Xi(i)
If the Permanent can be efficiently computed for most inputs, then it can for all inputs !
If the Permanent is hard in the worst-case, then it is also hard on average
Worst-case Average case reductionWorks for any low degree polynomial.Arithmetization: Boolean
functionspolynomials
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Back to Interactive Proofs
Thm [Nisan] Permanent 2IP
Thm [Lund-Fortnow-Karloff-Nisan] coNP IP
Thm [Shamir] IP = PSPACE
Thm [Babai-Fortnow-Lund] 2IP = NEXP
Thm [Arora-Safra, Arora-Lund-Motwani-Sudan-Szegedy] PCP = NP
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Conceptual meaning
Thm [Lund-Fortnow-Karloff-Nisan] coNP IP
- Tautologies have short, efficient proofs- can be replaced by (and prob interaction)
Thm [Shamir] IP = PSPACE
- Optimal strategies in games have efficient pfs- Optimal strategy of one player can simply be a
random strategy
Thm [Babai-Fortnow-Lund] 2IP = NEXP
Thm [Feige-Goldwasser-Lovasz-Safra-Szegedy]2IP=NEXP CLIQUE is hard to approximate
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PCP: Probabilistically Checkable Proofs
Prover Verifier
w
Wiles Referee
x
NP verifier: Can read all of wPCP verifier: Can (randomly) access
only a constant number of bits in
wPCP Thm [AS,ALMSS]: NP = PCPProofs robust proofs reductionPCP Thm [Dinur ‘06]: gap
amplification
WHPAccept (x,w)iff xLAccept (x,wi,wj,wk)iff xL
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3-SAT is 7/8+ hard
3-SAT is .99 hard
3-SAT is .995 hard
3-SAT is 1-1/n hard
IP NP3-SAT hard
2IP
PCP
3-SAT is 1-2/n hard
3-SAT is 1-4/n hard
InapproxGapamplification
SL=L SpaceComplexity
Algebraic
Com
bin
ato
rial
SL=L : Graph Connectivity LogSpace
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Getting out of mazes / Navigating unknown terrains (without map & memory)
Theseus
Ariadne
Thm [Aleliunas-Karp-Lipton-Lovasz-Rackoff ’80] A random walk will visit every vertex in n2 steps (with probability >99% )
Only a local view (logspace)n–vertex maze/graph
Thm [Reingold ‘05] : SL=L:A deterministic walk, computable in Logspace, will visit every vertex. Uses ZigZag expanders [Reingold-Vadhan-Wigderson]
Crete1000BC
Mars2006
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3-SAT is 1-4/n hard
3-SAT is .995 hard
3-SAT is .99 hard
3-SAT is 1-2/n hard
3-SAT is 1-1/n hard
3-SAT is hard
< 1-4/n
<.995
Expanders < .99
< 1-2/n
< 1-1/n
Connected graphs
Dinur’s proof ofPCP thm ‘06log n phases:Inapprox gapAmplificationPhase:Graph poweringand composition
(of the graphof constraints)
Reingold’s proofof SL=L ‘05log n phases:Spectral gapamplificationPhase:Graph poweringand composition
Expander graphs Comm Networks Error correction Complexity, Math …
Zigzagproduct
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Conclusions &Open Problems
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Computational definitions and reductionsgive new insights to millennia-old concepts,
e.g.Learning, Proof, Randomness, Knowledge,
Secret
Theorem(s): If FACTORING is hard then- Learning simple functions is impossible- Natural Proofs of hardness are impossible- Randomness does not speed-up algorithms- Zero-Knowledge proofs are possible- Cryptography & E-commerce are possible
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Can we base cryptography on PNP?Is 3-SAT hard on average?
Thm [Feigenbaum-Fortnow, Bogdanov-Trevisan]:
Not with Black-Box reductions, unless… The polynomial time hierarchy collapses.
Explore non Black-Box reductions!What can be gleaned from program code
?
Open Problems