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

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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
• (x1?x3 ?x7) (x1?x2 ?x4) (x5?x3 ?x6)
• 3-COLORING Find a vertex coloring
• using which
• satisfies all constraints

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Classic reductions

• ThmCook, Levin 71 3-SAT is NP-complete
• ThmKarp 72 3-COLORING is NP-complete
• Proof If 3-COLORING easy then 3-SAT easy
• formula ? graph
• satisfying ? legal
• assignment coloring
• Claim In every legal
• 3-coloring, z x ? y

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 II

• AI, Programming languages, Compilers,
  Verification, Scheduling,
• 3-SAT Find a Boolean assignment to the xi which
  satisfies most constraints
• (x1?x3 ?x7) (x1?x2 ?x4) (x5?x3 ?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

• ThmHastad 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-Szege
  dy, 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
2 Decades
Decade
NP 3-SAT is hard
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How to minimize players influence
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?
Function Influence Parity 1
Majority 1/7
Iterated Majority 1/8
Thm KahnKalaiLinial For every Boolean
function on n bits, some player has influence gt
(log n)/n (uses Functional Harmonic Analysis)
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3-SAT is 7/8? hard
IP Interactive Proofs 2IP 2-prover
Interactive Proofs PCP Probabilistically
Checkable Proofs

3-SAT is .99 hard
Probabilistic Interactive Proof systems
PCP
PCP NP
IP
NP 3-SAT hard
2IP
IP PSPACE
2IP NEXP
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IP (Probabilistic) Interactive Proofs Babai,
Goldwasser-Micali-Rackoff
x
Prover
Verifier
Accept (x,w) iff x?L
w
L ? NP
Wiles
Referee
Prover
q1
Verifier (probabilistic)
a1
L ? IP
qr
Accept (x,q,a) whp iff x?L
Socrates
Student
ar
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ZK IP Zero-Knowledge Interactive
Proofs Goldwasser-Micali-Rackoff
x
Prover
q1
Verifier
a1
L ?ZK IP
Accept (x,q,a) whp iff x?L
qr
Gets no other Information !
Socrates
Student
ar
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-Goldwasse
r-Kilian-Wigderson
x
Prover1
q1
Verifier
p1
Prover2
a1
b1
qr
pr
Socrates
Student
Plato
ar
br
Verif. accept (x,q,a,p,b) whp iff x?L
L?2IP
Theorem BGKW NP ? ZK 2IP
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What is the power of Randomness and Interaction
in Proofs?
Trivial inclusions IP ? PSPACE 2IP ?
NEXP Few nontrivial examples Graph non
isomorphism Few years of stalemate
2IP
Polynomial Space Nondeterministic Exponen
tial Time
IP
NP
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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 ? F and
input x, compute g(x) correctly when P errs on ?
(unknown) 1/8 of Fn
Easy case g is linear g(wz)g(w)g(z) Given x,
Pick y at random, and compute
P(xy)-P(y)
?x Proby P(xy)-P(y) ? g(x) ? 1/4
Fn
Other functions? - The Permanent BF,L - Low
degree polynomials
x
P errs
xy
y
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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) ThmGauss Determinant is
easy ThmValiant Permanent is hard
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The Permanent is self-correctable Beaver-Feigenba
um, Lipton
g(X) ???Sn ?i?n Xi?(i)
P errs on ?1/(8n)
Given X, pick Y at random, and compute P(X1Y),P(X
2Y), P(X(n1)Y)
WHP g(X1Y),g(X2Y), g(X(n1)Y) But
g(XuY)h(u) is a polynomial of deg
n. Interpolate to get h(0)g(X)



Fnxn
P errs
x
xy
x2y
x3y
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Hardness amplification

• X(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 reduction
• Works for any low degree polynomial.
• Arithmetization Boolean functions?polynomials

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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
• 2IPNEXP ? CLIQUE is hard to approximate

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PCP Probabilistically Checkable Proofs
x
Prover
Verifier
Accept (x,w) iff x?L
Accept (x,wi,wj,wk) iff x?L
w
WHP
Wiles
Referee
NP verifier Can read all of w PCP verifier Can
(randomly) access only a constant
number of bits in w PCP Thm AS,ALMSS NP
PCP Proofs ? robust proofs reduction PCP Thm
Dinur 06 gap amplification
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SLL Graph Connectivity ? LogSpace
3-SAT is 7/8? hard
Inapprox Gap amplification
SLL
3-SAT is .99 hard
3-SAT is .995 hard
3-SAT is 1-4/n hard
3-SAT is 1-2/n hard
PCP
Combinatorial
3-SAT is 1-1/n hard
IP
NP 3-SAT hard
2IP
Algebraic
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Getting out of mazes / Navigating unknown
terrains (without map memory)
Only a local view (logspace)
nvertex maze/graph
Mars 2006
Crete 1000BC
Thm Reingold 05 SLL A deterministic walk,
computable in Logspace, will visit every
vertex. Uses ZigZag expanders Reingold-Vadhan-W
igderson
Thm Aleliunas-Karp-Lipton-Lovasz-Rackoff 80 A
random walk will visit every vertex in n2 steps
(with probability gt99 )
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Dinurs proof of PCP thm 06 log n
phases Inapprox gap Amplification Phase Graph
powering and composition (of the graph of
constraints)
Reingolds proof of SLL 05 log n
phases Spectral gap amplification Phase Graph
powering and composition
3-SAT is hard
Connected graphs
3-SAT is 1-1/n hard
?lt 1-1/n
3-SAT is 1-2/n hard
?lt 1-2/n
3-SAT is 1-4/n hard
?lt 1-4/n
Zigzag product
3-SAT is .995 hard
?lt.995
3-SAT is .99 hard
Expanders ? lt .99
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• Conclusions
• Open Problems

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• Computational definitions and reductions
• give 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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Open Problems

• Can we base cryptography on P?NP?
• 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 ?