Wonders of the Digital Envelope

1
Happy Birthday Les !
2
Valiants Permanent gift to
TCS
to TCS

• Avi Wigderson
• Institute for Advanced Study

3
Valiants gift to me

• -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 algorithm
• OPEN Det NC alg for perfect matching.

4
The Permanent

• 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
X Pern(X) ???Sn ?i?n
Xi?(i)
5

• 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 the
• Permanent and friends
• Determinant, Perfect matching, counting

6
Monotone formulae for Majority
1
0
mk10
Valiant s random! Pr Fs ? Majk lt
exp(-k) OPEN Explicit? AKS, Determine m
(k2ltmltk5.3)
7
Counting classes PP, P, PP,
Gill PP
C(000) C(001) C(111)
C C(Z1,Z2,,Zn) is a small circuit/formula,
k2n,
Valiant P
C(000) C(001) C(111)
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The richness of P-complete problems
SAT CLIQUE SAT CLIQUE Permanent 2-SAT Netw
ork Reliability Monomer-Dimer Ising, Potts,
Tutte Enumeration, Algebra, Probability, Stat.
Physics
NP
C(000) C(001) C(111)

P
C(000) C(001) C(111)
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The power of counting Todas Theorem
PH P ? NP PSPACE
PP Valiant-Vazirani Poly-time
reduction C ? D OPEN Deterministic Valiant-Vazi
rani?
?
?
?
?
?
PROBABILISTIC
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Nice properties of Permanent Per 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
C errs on ?1/(8n) Interpolate Pern(X) from
C(XiY) with Y random, i1,2,,n1
Fnxn
C errs
x
xy
x2y
x3y
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Hardness amplification

• 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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Avalanche of consequencesto 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?

• EXP
• PSPACE Low degree extensions
• P Permenent
• PH
• NP No Black-Box reductions
• P Fortnow-Feigenbaum,Bogdanov-Trev
  isan
• NC2 Determinant
• L
• NC1 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 a of all matrices in Mn(Fp). a 1
? P BPP a 1-1/n ? P BPP Lipton a
1/nc ? P BPP CaiPavanSivakumar a n3/vp
? P PH AM FeigeLund a 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? Were done
EXP ? P/poly ? Per is EXP-complete Karp-Lipton,To
da workRSRDSRwork
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Non-relativizing Non-natural circuit lower
bounds
Vinodchandran PP ? SIZE(n10) Aaronson
This result doesnt relativize
Vinodchandrans Proof
PP ? P/poly? Were done
PP ? P/poly? PP MA LFKN? PP PP? ?2P ?
PP Toda? PP ? SIZE(n10) Kannan
Santhanam MA/1 ? SIZE(n10) OPEN Prove
NP ? SIZE(n10) Aaronson-Wigderson requires
non-algebrizing proofs
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The power of negation Arithmetic circuits
PMP(G) Perfect Matching polynomial of
G ShamirSnir,TiwariTompa msize(PMP(Kn,n)) gt
exp(n) FisherKasteleynTemperlysize(PMP(Gridn,n)
) poly(n) Valiant
msize(PMP(Gridn,n)) gt exp(n)
Boolean circuits
PM Perfect Matching function Edmonds
size(PM) poly(n) Razborov msize(PM) gt
nlogn OPEN tight? RazWigderson
mFsize(PM) gt exp(n)
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The power of Determinant (and linear algebra)
X?Mk(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)
nlogd
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Algebraic analog of P?NP

• F field, char(F)?2.
• X?Mk(F) Detk(X) ???Sk sgn(?) ?i?k Xi?(i)
• Y?Mn(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) lt exp(n)
• Mignon-Ressayre ?F k(n) gt n2
• Valiant k(n) ? poly(n) ?
  P?NP
• Mulmuley-Sohoni Algebraic-geometric approach

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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 nn 0/1 matrix.
• B Bij ? Aij at random
• Godsil-Gutman Pern(A) EDetn(B)2
• KarmarkarKarpLiptonLovaszLuby variance 2n
• B Bij ? AijRij with random Rij, ER0, ER21
• Use R?,?2,?31. variance 2n/2
• Chien-Rasmussen-Sinclair R non commutative!
• Use RC1,C2,..Cn elements of Clifford algebra.
• 
  variance poly(n)
• Approx scheme? OPEN Compute Det(B)

?
?
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Approx Pern deterministically

• A nn 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 Vardens conj)
• (2) Strongly polynomial algorithm for the
  following reduction to DS matrices
• Matrix scaling Find diagonal X,Y s.t. XAY is DS
• OPEN Find a deterministic subexp approx.

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Many happy returns, Les !!!