Happy Birthday Les Valiants Permanent gift to TCS
Happy Birthday Les !
Valiant’s Permanent gift to TCS Avi Wigderson Institute for Advanced Study
![Valiant’s gift to me -my postdoc problems! [Valiant ’ 82] “Parallel computation”, Proc. Of](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-3.jpg "Valiant’s gift to me -my postdoc problems! [Valiant ’ 82] “Parallel computation”, Proc. Of")
Valiant’s gift to me -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.
The Permanent X 11, X 12, …, X 1 n X= X 21, X 22, …, X 2 n … … Xn 1, Xn 2, …, Xnn to TCS Pern(X) = Sn i [n] Xi (i) [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 ½ an hour about the Permanent and friends: Determinant, Perfect matching, counting
Monotone formulae for Majority M X Y 17 1 Y 2 X 3 X Y 37 1 Xk 1 0 X 1 V V V σ X 2 V V F 0 X Ym 1 X 2 V X 1 m=k 10 V [Valiant]: σ random! Pr[ Fσ ≠ Majk ] < exp(-k) OPEN: Explicit? [AKS], Determine m (k 2<m<k 5. 3)
![Counting classes: PP, #P, P#P, … M [Gill] PP X 1 X 2 X](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-7.jpg "Counting classes: PP, #P, P#P, … M [Gill] PP X 1 X 2 X")
Counting classes: PP, #P, P#P, … M [Gill] PP X 1 X 2 X 3 C(00… 0) C(00… 1) … … Xk C(11… 1) C = C(Z 1, Z 2, …, Zn) is a small circuit/formula, k=2 n, + [Valiant] #P X 1 X 2 X 3 C(00… 0) C(00… 1) … … Xk C(11… 1)
The richness of #P-complete problems SAT CLIQUE NP V C(00… 0) C(00… 1) … … #SAT #CLIQUE Permanent #2 -SAT Network Reliability Monomer-Dimer Ising, Potts, Tutte #P C(00… 0) C(00… 1) … … C(11… 1) + C(11… 1) Enumeration, Algebra, Probability, Stat. Physics
![The power of counting: Toda’s Theorem P NP PH P#P [Valiant-Vazirani] PROBABILISTIC Poly-time reduction:](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-9.jpg "The power of counting: Toda’s Theorem P NP PH P#P [Valiant-Vazirani] PROBABILISTIC Poly-time reduction:")
The power of counting: Toda’s Theorem P NP PH P#P [Valiant-Vazirani] PROBABILISTIC Poly-time reduction: C D OPEN: Deterministic Valiant-Vazirani? PSPACE NP V C(00… 0) C(00… 1) … … + P D(00… 0) C(00… 1) … … C(11… 1) + C(11… 1)
![Nice properties of Permanent Per is downwards self-reducible Pern(X) = Sn i [n] Xi](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-10.jpg "Nice properties of Permanent Per is downwards self-reducible Pern(X) = Sn i [n] Xi")
Nice properties of Permanent Per is downwards self-reducible Pern(X) = Sn i [n] Xi (i) Pern(X) = i [n] Pern-1(X 1 i) Per is random self-reducible [Beaver-Feigenbaum, Lipton] C errs on 1/(8 n) Interpolate Pern(X) from C(X+i. Y) with Y random, i=1, 2, …, n+1 C errs x+3 y x+2 y 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 functions polynomials
Avalanche of consequences to probabilistic proof systems Using both RSR and DSR of Permanent! [Nisan] Per 2 IP [Lund-Fortnow-Karloff-Nisan] Per IP [Shamir] IP = PSPACE [Babai-Fortnow-Lund] 2 IP = NEXP [Arora-Safra, Arora-Lund-Motwani-Sudan-Szegedy] PCP = NP
Which classes have complete RSR problems? EXP PSPACE #P PH NP P NC 2 L NC 1 Low degree extensions Permenent No Black-Box reductions ? [Fortnow-Feigenbaum, Bogdanov-Trevisan] Determinant [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 α of all matrices in Mn(Fp). α =1 #P = BPP α =1 -1/n #P = BPP [Lipton] α =1/nc #P = BPP [Cai. Pavan. Sivakumar] α =n 3/√p #P = PH =AM [Feige. Lund] α =1/p possible! OPEN: Tighten the bounds! (Improve Reed-Solomon list decoding [Sudan, …])
![Hardness vs. Randomness [Babai-Fortnaow-Nisan-Wigderson] EXP P/poly BPP SUBEXP [Impagliazzo-Wigderson] EXP ≠ BPP SUBEXP Proof:](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-15.jpg "Hardness vs. Randomness [Babai-Fortnaow-Nisan-Wigderson] EXP P/poly BPP SUBEXP [Impagliazzo-Wigderson] EXP ≠ BPP SUBEXP Proof:")
Hardness vs. Randomness [Babai-Fortnaow-Nisan-Wigderson] EXP P/poly BPP SUBEXP [Impagliazzo-Wigderson] EXP ≠ BPP SUBEXP Proof: EXP P/poly We’re done EXP P/poly Per is EXP-complete [Karp-Lipton, Toda] …work…RSR…DSR…work… [Kabanets-Impagliazzo] Permanent is easy iff Identity Testing can be derandomized
![Non-relativizing & Non-natural circuit lower bounds [Vinodchandran]: PP SIZE(n 10) [Aaronson]: This result doesn’t](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-16.jpg "Non-relativizing & Non-natural circuit lower bounds [Vinodchandran]: PP SIZE(n 10) [Aaronson]: This result doesn’t")
Non-relativizing & Non-natural circuit lower bounds [Vinodchandran]: PP SIZE(n 10) [Aaronson]: This result doesn’t relativize Vinodchandran’s Proof: PP P/poly We’re done Non-Relativizing PP P/poly P#P = MA [LFKN] Non-Natural P#P = PP 2 P PP [Toda] PP SIZE(n 10) [Kannan] [Santhanam]: MA/1 SIZE(n 10) OPEN: Prove NP SIZE(n 10) [Aaronson-Wigderson] requires non-algebrizing proofs
The power of negation Arithmetic circuits PMP(G) – Perfect Matching polynomial of G [Shamir. Snir, Tiwari. Tompa]: msize(PMP(Kn, n)) > exp(n) [Fisher. Kasteleyn. Temperly]: 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 [Raz. Wigderson]: m. Fsize(PM) > exp(n) OPEN: tight?
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 [Fisher. Kasteleyn. Temperly]: counting perfect matchings in planar n-graphs ≤ Detn [Valiant, Cai-Lu] Holographic algorithms … [Valiant]: evaluating size n formulae ≤ Detn [Hyafill, Valiant. Skyum. Berkowitz. Rackoff]: evaluating size n degree d arithmetic circuits OPEN: Improve to Detpoly(n, d) ≤ Detnlogd
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 a b Per 2 = Det 2 c d -c d [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
![Detn vs. Pern [Nisan] Both require noncommutative arithmetic branching programs of size 2 n](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-20.jpg "Detn vs. Pern [Nisan] Both require noncommutative arithmetic branching programs of size 2 n")
Detn vs. Pern [Nisan] Both require noncommutative arithmetic branching programs of size 2 n [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 n 22 n OPEN: Improve n! for Detn
![Approximating Pern A: n×n 0/1 matrix. B: Bij ±Aij at random [Godsil-Gutman] Pern(A) =](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-21.jpg "Approximating Pern A: n×n 0/1 matrix. B: Bij ±Aij at random [Godsil-Gutman] Pern(A) =")
Approximating Pern A: n×n 0/1 matrix. B: Bij ±Aij at random [Godsil-Gutman] Pern(A) = E[Detn(B)2] [Karmarkar. Karp. Lipton. Lovasz. Luby] variance = 2 n… B: Bij Aij. Rij with random Rij, E[R]=0, E[R 2]=1 Use R={ω, ω2, ω3=1}. variance ≤ 2 n/2 [Chien-Rasmussen-Sinclair] R non commutative! Use R={C 1, C 2, . . Cn} elements of Clifford algebra. variance ≤ poly(n) Approx scheme? OPEN: Compute Det(B)
![Approx Pern deterministically A: n×n non-negative real matrix. [Linial-Samorodnitsky-Wigderson] Deterministic e-n -factor approximation. Two](http://slidetodoc.com/presentation_image_h/2a962f2af53ea53cffb95f1931ef333d/image-22.jpg "Approx Pern deterministically A: n×n non-negative real matrix. [Linial-Samorodnitsky-Wigderson] Deterministic e-n -factor approximation. Two")
Approx Pern 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!/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.
Many happy returns, Les !!!
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