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Wonders of the Digital Envelope

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Of 7th IBM symposium on mathematical foundations of computer science. ... Works for any low degree polynomial. Arithmetization: Boolean functions ... – PowerPoint PPT presentation

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Title: 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)
8
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)
9
The power of counting Todas Theorem
PH P ? NP PSPACE
PP Valiant-Vazirani Poly-time
reduction C ? D OPEN Deterministic Valiant-Vazi
rani?
?
?
?
?
?
PROBABILISTIC
10
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
11
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

12
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

13
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?

?
14
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,)
15
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
16
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
17
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)
18
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
19
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

20
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

21
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-Luby-Rassmusen R non commutative!
  • Use RC1,C2,..Cn elements of Clifford algebra.

  • variance poly(n)
  • Approx scheme? OPEN Compute Det(B)

?
?
22
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.

23
Many happy returns, Les !!!
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