1. Happy Birthday Les !

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

3. -my postdoc problems! [Valiant ’82] “Parallel computation”, Proc. Of 7 th 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 X = Per n (X) =   Sn  i  [n] X i  (i) X 11,X 12,…, X 1n X 21,X 22,…, X 2n … … … … X n1,X n2,…, X nn [Valiant ’79] “The complexity of computing the permanent” [Valiant ‘79] “The complexity of enumeration and reliability problems” to TCS

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 [Valiant]: σ random! Pr[ F σ ≠ Maj k ] < exp(-k) OPEN: Explicit? [AKS], Determine m (k 2 <m<k 5.3 ) M X1X1 X2X2 X3X3 XkXk Y1Y1 Y2Y2 Y3Y3 YmYm V V V V V V V F 10 m=k 10 σ X7X7 1X7X7 X1X1 V V V V V V V F 1X2X2 X1X1 0

7. Counting classes: PP, #P, P #P, … C = C(Z 1,Z 2,…,Z n ) is a small circuit/formula, k=2 n, M X1X1 X2X2 X3X3 XkXk + X1X1 X2X2 X3X3 XkXk C(00…0) C(00…1) … … C(11…1) [Gill] PP [Valiant] #P C(00…0) C(00…1) … … C(11…1)

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

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 C(00…0) C(00…1) … … C(11…1) NP   + P D(00…0) C(00…1) … … C(11…1) + PROBABILISTIC

10. Nice properties of Permanent Per is downwards self-reducible Per n (X) =   Sn  i  [n] X i  (i) Per n (X) =  i  [n] Per n-1 (X 1i ) Per is random self-reducible [Beaver-Feigenbaum, Lipton] F nxn C errs x+3yx+3y x+2yx+2y x x+y C errs on  1/(8n) Interpolate Per n (X) from C(X+iY) with Y random, i=1,2,…,n+1

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 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

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] NC 2 Determinant L NC 1 [Barrington] OPEN: Non Black-Box reductions? ?

14. On what fraction of inputs can we compute Permanent? Assume: a PPT algorithm A computer Per n for on fraction α of all matrices in M n (F p ). α = 1  #P = BPP α = 1-1/n  #P = BPP [Lipton] α = 1/n c  #P = BPP [CaiPavanSivakumar] α = n 3 /√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. [Vinodchandran]: PP  SIZE(n 10 ) [Aaronson]: This result doesn’t relativize [Santhanam]: MA /1  SIZE(n 10 ) OPEN: Prove NP  SIZE(n 10 ) [Aaronson-Wigderson] requires non-algebrizing proofs Vinodchandran’s Proof: PP  P/poly  We’re done PP  P/poly  P #P = MA [LFKN]  P #P = PP   2 P  PP [Toda]  PP  SIZE(n 10 ) [Kannan] Non-Relativizing Non-Natural Non-relativizing & Non-natural circuit lower bounds

17. PMP(G) – Perfect Matching polynomial of G [ShamirSnir,TiwariTompa]: msize(PMP(K n,n )) > exp(n) [FisherKasteleynTemperly]:size(PMP(Grid n,n )) = poly(n) [Valiant]: msize(PMP(Grid n,n )) > exp(n) The power of negation Arithmetic circuits Boolean circuits PM – Perfect Matching function [Edmonds]: size(PM) = poly(n) [Razborov]: msize(PM) > n logn OPEN: tight? [RazWigderson]: mFsize(PM) > exp(n)

18. X  M k (F) Det k (X) =   Sk sgn(  )  i  [k] X i  (i) [Kirchoff]: counting spanning trees in n-graphs ≤ Det n [FisherKasteleynTemperly]: counting perfect matchings in planar n-graphs ≤ Det n [Valiant, Cai-Lu] Holographic algorithms … [Valiant]: evaluating size n formulae ≤ Det n [Hyafill, ValiantSkyumBerkowitzRackoff]: evaluating size n degree d arithmetic circuits ≤ Det OPEN: Improve to Det poly(n,d) The power of Determinant (and linear algebra) n logd

19. Algebraic analog of “P  NP” F field, char(F)  2. X  M k (F) Det k (X) =   Sk sgn(  )  i  [k] X i  (i) Y  M n (F) Per n (Y) =   Sn  i  [n] Y i  (i) Affine map L: M n (F)  M k (F) is good if Per n = Det k  L k(n): the smallest k for which there is a good map? [Polya] k(2) =2 Per 2 = Det 2 [Valiant]  F k(n) < exp(n) [Mignon-Ressayre]  F k(n) > n 2 [Valiant] k(n)  poly(n)  “P  NP” [Mulmuley-Sohoni] Algebraic-geometric approach a b -c d a b c d

20. Det n vs. Per n [Nisan] Both require noncommutative arithmetic branching programs of size 2 n [Raz] Both require multilinear arithmetic formulae of size n logn [Pauli,Troyansky-Tishby] Both equally computable by nature- quantum state of n identical particles: bosons  Per n, fermions  Det n [Ryser] Per n has depth-3 circuits of size n 2 2 n OPEN: Improve n! for Det n

21. Approximating Per n A: n×n 0/1 matrix. B: B ij  ± A ij at random [Godsil-Gutman] Per n (A) = E[Det n (B) 2 ] [KarmarkarKarpLiptonLovaszLuby] variance = 2 n … B: B ij  A ij R ij with random R ij, E[R]=0, E[R 2 ]=1 Use R={ ω,ω 2,ω 3 = 1}. variance ≤ 2 n/2 [Chien-Rasmussen-Sinclair] R non commutative! Use R={C 1,C 2,..C n } elements of Clifford algebra. variance ≤ poly(n) Approx scheme? OPEN: Compute Det(B) 

22. Approx Per n deterministically A: n×n 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!/n n ≤ 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 !!!