Happy Birthday Les !

Valiant’s Permanent gift to TCS

Avi WigdersonInstitute for Advanced

Study

to TCS

-my postdoc problems![Valiant ’82] “Parallel computation”, Proc. Of 7th IBM symposium on mathematical foundations of computer science.Are the following “inherently sequential”?-Finding maximal independent set?[Karp-Wigderson] No! NC algorithm. -Finding a perfect matching?[Karp-Upfal-Wigderson] No! RNC algorithmOPEN: Det NC alg for perfect matching.

Valiant’s gift to me

The Permanent

X = Pern(X) = Sn i[n] Xi(i)

X11,X12,…, X1n

X21,X22,…, X2n

… … … … Xn1,Xn2,…, Xnn

[Valiant ’79] “The complexity of computing the permanent”[Valiant ‘79] “The complexity of enumeration and reliability problems”

to TCS

Valiant brought the Permanent, polynomials and Algebra into the focus of TCS research.

Plan of the talk As many results and questions as I can squeeze in ½ an hour about thePermanent and friends:Determinant, Perfect matching, counting

Monotone formulae for Majority

[Valiant]: σ random! Pr[ Fσ ≠ Majk ] < exp(-k)

OPEN: Explicit? [AKS], Determine m (k2<m<k5.3)

M

X1 X2 X3 Xk

Y1 Y2 Y3 Ym

V

V

VV

V VV

F

1 0

m=k10

σ

X7 1 X7 X1

V

V

VV

V VV

F

1 X2 X1 0

Counting classes: PP, #P, P#P, …

C = C(Z1,Z2,…,Zn) is a small circuit/formula, k=2n,

M

X1 X2 X3 Xk

+

X1 X2 X3 Xk

C(00…0) C(00…1) … … C(11…1)

[Gill] PP

[Valiant] #P

C(00…0) C(00…1) … … C(11…1)

The richness of #P-complete problems V

+

C(00…0) C(00…1) … … C(11…1)

NP

#P

C(00…0) C(00…1) … … C(11…1)

SATCLIQUE

#SAT#CLIQUEPermanent#2-SATNetwork ReliabilityMonomer-DimerIsing, Potts, TutteEnumeration, Algebra, Probability, Stat. Physics

The power of counting: Toda’s Theorem

PHP NP PSPACE P#P

[Valiant-Vazirani] Poly-time reduction:C D

OPEN: DeterministicValiant-Vazirani?

V

C(00…0) C(00…1) … … C(11…1)

NP

+P

D(00…0) C(00…1) … … C(11…1)

+

PROBABILISTIC

Nice properties of PermanentPer is downwards self-reducible

Pern(X) = Sn i[n] Xi(i)

Pern(X) = i[n] Pern-1(X1i)

Per is random self-reducible[Beaver-Feigenbaum, Lipton]

Fnxn

C errs

x+3yx+2y

xx+y

C errs on 1/(8n)Interpolate Pern(X)

from C(X+iY) with Y random, i=1,2,…,n+1

Hardness amplificationIf 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

functionspolynomials

Avalanche of consequences

to probabilistic proof systems

Using both RSR and DSR of Permanent!

[Nisan] Per 2IP

[Lund-Fortnow-Karloff-Nisan] Per IP

[Shamir] IP = PSPACE

[Babai-Fortnow-Lund] 2IP = NEXP

[Arora-Safra,Arora-Lund-Motwani-Sudan-Szegedy] PCP

= NP

Which classes have complete RSR problems?

EXPPSPACE Low degree extensions#P PermenentPHNP No Black-Box reductionsP [Fortnow-Feigenbaum,Bogdanov-

Trevisan] NC2 DeterminantLNC1 [Barrington]

OPEN: Non Black-Box reductions?

?

On what fraction of inputs can we compute Permanent?

Assume: a PPT algorithm A computer Pern for on fraction α of all matrices in Mn(Fp).

α =1 #P = BPPα =1-1/n #P = BPP [Lipton]α =1/nc #P = BPP [CaiPavanSivakumar]α =n3/√p #P = PH =AM [FeigeLund]α =1/p possible!

OPEN: Tighten the bounds!(Improve Reed-Solomon list decoding [Sudan,…])

Hardness vs. Randomness

[Babai-Fortnaow-Nisan-Wigderson]EXP P/poly BPP SUBEXP

[Impagliazzo-Wigderson]EXP ≠ BPP BPP SUBEXP

[Kabanets-Impagliazzo] Permanent is easy iff Identity Testing can be derandomized

Proof:

EXP P/poly We’re done

EXP P/poly Per is EXP-complete

[Karp-Lipton,Toda]…work…RSR…DSR…

work…

[Vinodchandran]: PP SIZE(n10) [Aaronson]: This result doesn’t relativize

[Santhanam]: MA/1 SIZE(n10)

OPEN: Prove NP SIZE(n10) [Aaronson-Wigderson] requires non-algebrizing proofs

Vinodchandran’s Proof:

PP P/poly We’re

done

PP P/poly P#P = MA [LFKN]

P#P = PP 2P PP [Toda] PP SIZE(n10)

[Kannan]

Non-Relativizing

Non-Natural

Non-relativizing & Non-natural

circuit lower bounds

PMP(G) – Perfect Matching polynomial of G

[ShamirSnir,TiwariTompa]: msize(PMP(Kn,n)) > exp(n)

[FisherKasteleynTemperly]:size(PMP(Gridn,n)) = poly(n)

[Valiant]: msize(PMP(Gridn,n)) > exp(n)

The power of negation Arithmetic circuits

Boolean circuitsPM – Perfect Matching function

[Edmonds]: size(PM) = poly(n)

[Razborov]: msize(PM) > nlogn OPEN: tight?

[RazWigderson]: mFsize(PM) > exp(n)

XMk(F) Detk(X) = Sk sgn() i[k] Xi(i)

[Kirchoff]: counting spanning trees in n-graphs ≤ Detn

[FisherKasteleynTemperly]:

counting perfect matchings in planar n-graphs ≤ Detn

[Valiant, Cai-Lu] Holographic algorithms …

[Valiant]: evaluating size n formulae ≤ Detn

[Hyafill, ValiantSkyumBerkowitzRackoff]: evaluating

size n degree d arithmetic circuits ≤ Det

OPEN: Improve to Detpoly(n,d)

The power of Determinant(and linear algebra)

nlogd

Algebraic analog of “PNP”

F field, char(F)2.XMk(F) Detk(X) = Sk sgn() i[k] Xi(i)

YMn(F) Pern(Y) = Sn i[n] Yi(i)

Affine map L: Mn(F) Mk(F) is good if Pern = Detk L

k(n): the smallest k for which there is a good map?

[Polya] k(2) =2 Per2 = Det2

[Valiant] F k(n) < exp(n)[Mignon-Ressayre] F k(n) > n2

[Valiant] k(n) poly(n) “PNP”[Mulmuley-Sohoni] Algebraic-geometric approach

a b-c d

a bc d

Detn vs. Pern

[Nisan] Both require noncommutative arithmetic branching programs of size 2n

[Raz] Both require multilinear arithmetic formulae of size nlogn

[Pauli,Troyansky-Tishby] Both equally computable by nature- quantum state of n identical particles: bosons Pern, fermions Detn

[Ryser] Pern has depth-3 circuits of size n22n

OPEN: Improve n! for Detn

Approximating Pern

A: n×n 0/1 matrix. B: Bij ±Aij at random

[Godsil-Gutman] Pern(A) = E[Detn(B)2]

[KarmarkarKarpLiptonLovaszLuby] variance = 2n…B: Bij AijRij with random Rij, E[R]=0, E[R2]=1

Use R={ω,ω2,ω3=1}. variance ≤ 2n/2

[Chien-Luby-Rassmusen] R non commutative!Use R={C1,C2,..Cn} elements of Clifford algebra.

variance ≤ poly(n)

Approx scheme? OPEN: Compute Det(B)

Approx Pern

deterministicallyA: n×n non-negative real matrix. [Linial-Samorodnitsky-Wigderson]Deterministic e-n -factor approximation.Two ingredients:(1) [Falikman,Egorichev] If B Doubly Stochastic

then e-n ≈ n!/nn ≤ Per(B) ≤ 1(the lower bound solved van der Varden’s conj)(2) Strongly polynomial algorithm for the following reduction to DS matrices:Matrix scaling: Find diagonal X,Y s.t. XAY is DSOPEN: Find a deterministic subexp approx.

Many happy returns, Les !!!