happy birthday les !. valiant’s permanent gift to tcs avi wigderson institute for advanced study...
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![Page 1: Happy Birthday Les !. Valiant’s Permanent gift to TCS Avi Wigderson Institute for Advanced Study to TCS](https://reader035.vdocuments.site/reader035/viewer/2022062516/56649d3f5503460f94a188ff/html5/thumbnails/1.jpg)
Happy Birthday Les !
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Valiant’s Permanent gift to TCS
Avi WigdersonInstitute for Advanced
Study
to TCS
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-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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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?
?
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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,…])
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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…
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[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
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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)
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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
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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
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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
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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)
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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.
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Many happy returns, Les !!!