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Happy Birthday Les !

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.

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Happy Birthday Les !

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  1. Happy Birthday Les !

  2. to TCS Valiant’s Permanent gift to TCS Avi Wigderson Institute for Advanced Study

  3. -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. Valiant’s gift to me

  4. The Permanent to TCS 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” 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 M 1 0 σ X1 X2 X3 Xk Y1 X7 1 Y2 X7 Y3 X1 Ym 1 0 X1 X2 V V V V V V V V V V V V m=k10 V V F F [Valiant]: σ random!Pr[ Fσ ≠ Majk ] < exp(-k) OPEN: Explicit? [AKS], Determine m (k2<m<k5.3)

  7. Counting classes: PP, #P, P#P, … [Gill] PP M C(00…0) C(00…1) … …C(11…1) X1 X2 X3 Xk C = C(Z1,Z2,…,Zn) is a small circuit/formula, k=2n, + [Valiant] #P X1 X2 X3 Xk C(00…0) C(00…1) … …C(11…1)

  8. The richness of #P-complete problems SAT CLIQUE #SAT #CLIQUE Permanent #2-SAT Network Reliability Monomer-Dimer Ising, Potts, Tutte Enumeration, Algebra, Probability, Stat. Physics NP V C(00…0) C(00…1) … …C(11…1) #P + C(00…0) C(00…1) … …C(11…1)

  9. The power of counting: Toda’s Theorem PH P  NP PSPACE P#P [Valiant-Vazirani] Poly-time reduction: C  D OPEN: Deterministic Valiant-Vazirani?    V   NP PROBABILISTIC + C(00…0) C(00…1) … …C(11…1) P + D(00…0) C(00…1) … … C(11…1)

  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) InterpolatePern(X) from C(X+iY) with Y random, i=1,2,…,n+1 Fnxn C errs x x+y x+2y x+3y

  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-Trevisan] 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 α of all matrices in Mn(Fp). α =1 #P = BPP α =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,…])

  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 We’re done EXP  P/poly  Per is EXP-complete [Karp-Lipton,Toda] …work…RSR…DSR…work…

  16. Non-relativizing & Non-natural circuit lower bounds Non-Relativizing Non-Natural [Vinodchandran]: PP  SIZE(n10) [Aaronson]: This result doesn’t relativize 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] [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)) > exp(n) [FisherKasteleynTemperly]:size(PMP(Gridn,n)) = poly(n) [Valiant]: msize(PMP(Gridn,n)) > exp(n) Boolean circuits PM– Perfect Matching function [Edmonds]: size(PM) = poly(n) [Razborov]:msize(PM) > nlogn OPEN: tight? [RazWigderson]: mFsize(PM) > exp(n)

  18. The power of Determinant (and linear algebra) XMk(F) Detk(X) = Sksgn() 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 mapL: Mn(F)  Mk(F) is good if Pern = Detk L k(n): the smallest k for which there is a goodmap? [Polya] k(2) =2 Per2 = Det2 [Valiant] F k(n) < exp(n) [Mignon-Ressayre] Fk(n) > n2 [Valiant]k(n)  poly(n) “PNP” [Mulmuley-Sohoni] Algebraic-geometric approach a b c d a b -c d

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

  22. Approx Pern deterministically A: n×n non-negative real matrix. [Linial-Samorodnitsky-Wigderson] Deterministice-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 DS OPEN: Find a deterministic subexp approx.

  23. Many happy returns, Les !!!

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