1 / 23

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.

clark
Download Presentation

Happy Birthday Les !

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 !!!

More Related