the art of reduction (or, depth through breadth) avi wigderson institute for advanced study...
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The art of reduction
(or, Depth through Breadth)
Avi WigdersonInstitute 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
If A is easy then B is easy
If B is hard then A is hard
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
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
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.
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?
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
3-SAT is 7/8+ hard
3-SAT is .99 hard
NP3-SAT is hard
Influencesin Booleanfunctions
DistributedComputing
2 DecadesDecade
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?
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
Conclusions &Open Problems
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
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