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