# Wonders of the Digital Envelope

Happy Birthday Les ! Valiants Permanent gift to TCS to TCS Avi Wigderson Institute for Advanced 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? The Permanent X11,X12,, X1n X21,X22,, X2n X = X ,X ,, X Xi(i) n1 n2 nn to Pern(X) = Sn i[n] TCS

[Valiant 79] The complexity of computing the permanent [Valiant 79] The complexity of enumeration and reliability problems 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 an hour about the Permanent and friends: Determinant, Perfect matching, counting Monotone formulae for Majority M Y

X17 Y2 1 X3 Y X37 1 0 V V V X1

F 0 Y Xm1 X2 V V [Valiant]: random! exp(-k) 1 Xk V

X2 V X1 m=k10 Pr[ F Majk ] < Counting classes: PP, #P, P#P, M [Gill] PP X1 X2 X3

C(000) C(001) Xk C(111) C = C(Z1,Z2,,Zn) is a small circuit/formula, k=2n, + [Valiant] #P X1 X2 X3 C(000) C(001) Xk C(111)

The richness of #P-complete problems V NP SAT CLIQUE C(000) C(001) #SAT #P #CLIQUE Permanent C(000) C(001) #2-SAT Network Reliability Monomer-Dimer Ising, Potts, Tutte

C(111) + C(111) Enumeration, Algebra, Probability, Stat. The power of counting: Todas Theorem PH P NP PSPACE P#P NP [Valiant-Vazirani]

PROBABILISTIC Poly-time reduction:C(000) C(001) CD OPEN: Deterministic Valiant-Vazirani? +P D(000) C(001) V C(111) + C(111) 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(X+iY) with Y random, C errs x+2y x+3y x+y Fnxn

x 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 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? EXP PSPACE Low degree extensions #P Permenent PH NP No Black-Box reductions P [Fortnow-Feigenbaum,BogdanovTrevisan] NC2 Determinant

L NC1 [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 of all matrices in Mn(Fp). =1 #P = BPP =1-1/n #P = BPP [Lipton] =1/nc #P = BPP [CaiPavanSivakumar] =n3/p

#P = PH =AM [FeigeLund] =1/p possible! Hardness vs. Randomness [Babai-Fortnaow-Nisan-Wigderson] EXP P/poly BPP SUBEXP [Impagliazzo-Wigderson] EXP BPP BPP SUBEXP Proof: EXP P/poly Were done EXP P/poly

Per is EXP-complete [Karp-Lipton,Toda] workRSRDSR work [Kabanets-Impagliazzo] Permanent is easy iff Identity Testing can be derandomized Non-relativizing & Nonnatural circuit lower bounds [Vinodchandran]: PP SIZE(n10) [Aaronson]: This result doesnt relativize Vinodchandrans Proof: PP P/poly Were done [Santhanam]:

NonRelativizing PP P/poly P#P = MA [LFKN] Non-Natural P#P = PP 2P PP [Toda] PP SIZE(n10) [Kannan] MA/1 SIZE(n10) OPEN: Prove NP SIZE(n10) [Aaronson-Wigderson] requires non- The power of negation Arithmetic circuits PMP(G) Perfect Matching polynomial of G

[ShamirSnir,TiwariTompa]: msize(PMP(Kn,n)) > exp(n) [FisherKasteleynTemperly]:size(PMP(Gridn,n)) = poly(n) Boolean circuitsn,n)) > [Valiant]: msize(PMP(Grid PM Perfect Matching function exp(n) [Edmonds]: size(PM) = poly(n) [Razborov]: tight? msize(PM) > nlogn

OPEN: The power of Determinant (and linear algebra) 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]: nlogd

Algebraic analog of PNP F field, char(F)2. XMk(F) YMn(F) Detk(X) = Sk sgn() i[k] Xi(i) 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 a b = Det2 cd

ab -c d [Valiant] F k(n) < exp(n) [Mignon-Ressayre] F k(n) > n2 [Valiant] k(n) poly(n) PNP [Mulmuley-Sohoni] Algebraic-geometric approach 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: nn 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={,2,3=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 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. Many happy returns, Les !!!

## Recently Viewed Presentations

• and Brett Combs 2012 Elastic /Inelastic Measurement Project Critical need for high-precision and accurate elastic and inelastic neutron scattering data on materials important for fission reactor technology.
• True or False Quiz on Tithing. A tithe in OT Law was 10% of one's money or income. OT Law required Israelites to tithe 10% annually. 3. The NT letters mention tithing for Christians
• ESSA guidelines indicate that math flexibility is allowed when high school math courses are administered as part of the state's plan. Because Colorado is no longer administering the high school math assessments at high school (PSAT and SAT instead), this...
• GCSE - GCE - BTEC. Functional Skills. Edexcel Awards. White label - International GCSE and CiDA/DiDA qualifications. ASK: what do you do if you don't receive a label? Go to labels form on website (DEMO) Do not complete the online...
• The science and engineering of the materials, devices and systems that lie at the heart of this revolution have been major research areas at the University of Surrey for several decades. The Institute contains about 100 people and has a...
• The Latin word for priest is "pontifex," which means bridge builder. We will only appreciate our need for a high priest to the degree that we realize how holy and unapproachable God is and how sinful and defiled we are....
• Willow Creek (in excess of 2000 home sites) Infill of Plantation Oak. 16 new apartments on Knox McRae. Vacant unimproved property in many locations, already zoned multi-family. Construction Delays impact QOL. Plantation Oaks and the not yet started Country Club...
• My Mother is a Baker My Mother is a Baker My mother is a baker. My father is a trash man. My sister is a singer. My brother is a cowboy. My doggie is a licker. My kitty is a...