happy birthday les !

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Happy Birthday Les !

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Happy Birthday Les !. to TCS. Valiant’s Permanent gift to TCS. Avi Wigderson Institute for Advanced Study. -my postdoc problems! [Valiant ’82] “Parallel computation”, Proc. Of 7 th IBM symposium on mathematical foundations of computer science. - PowerPoint PPT Presentation

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Page 1: Happy Birthday      Les !

Happy Birthday Les !

Page 2: Happy Birthday      Les !

Valiant’s Permanent gift to TCS

Avi WigdersonInstitute for Advanced

Study

to TCS

Page 3: Happy Birthday      Les !

-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

Page 4: Happy Birthday      Les !

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

Page 5: Happy Birthday      Les !

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

Page 6: Happy Birthday      Les !

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

σX7 1 X7 X1

V

V

VV

V VV

F

1 X2 X1 0

Page 7: Happy Birthday      Les !

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)

Page 8: Happy Birthday      Les !

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

Page 9: Happy Birthday      Les !

The power of counting: Toda’s Theorem PH

P 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

Page 10: Happy Birthday      Les !

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+2yx

x+y

C errs on 1/(8n)Interpolate Pern(X)from C(X+iY) with Y random, i=1,2,…,n+1

Page 11: Happy Birthday      Les !

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

Page 12: Happy Birthday      Les !

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

Page 13: Happy Birthday      Les !

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?

?

Page 14: Happy Birthday      Les !

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,…])

Page 15: Happy Birthday      Les !

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…

Page 16: Happy Birthday      Les !

[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

Page 17: Happy Birthday      Les !

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)

Page 18: Happy Birthday      Les !

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]: evaluatingsize n degree d arithmetic circuits ≤ DetOPEN: Improve to Detpoly(n,d)

The power of Determinant(and linear algebra)

nlogd

Page 19: Happy Birthday      Les !

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 Lk(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

Page 20: Happy Birthday      Les !

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

Page 21: Happy Birthday      Les !

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]=1Use R={ω,ω2,ω3=1}. variance ≤ 2n/2

[Chien-Rasmussen-Sinclair] R non commutative!Use R={C1,C2,..Cn} elements of Clifford algebra. variance ≤ poly(n)Approx scheme? OPEN: Compute Det(B)

Page 22: Happy Birthday      Les !

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.

Page 23: Happy Birthday      Les !

Many happy returns, Les !!!