Happy Birthday Les !

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Happy Birthday Les !. to TCS. Valiants 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. - PowerPoint PPT Presentation

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<ul><li><p> Happy Birthday Les !</p></li><li><p>Valiants Permanent gift to TCS Avi WigdersonInstitute for Advanced Studyto TCS</p></li><li><p>-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 algorithmOPEN: Det NC alg for perfect matching.</p><p>Valiants gift to me </p></li><li><p>The PermanentX = Pern(X) = Sn i[n] Xi(i) </p><p> X11,X12,, X1n X21,X22,, X2n Xn1,Xn2,, Xnn</p><p>[Valiant 79] The complexity of computing the permanent[Valiant 79] The complexity of enumeration and reliability problems to TCS</p></li><li><p>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 thePermanent and friends:Determinant, Perfect matching, counting</p></li><li>Monotone formulae for Majority[Valiant]: random! Pr[ F Majk ] &lt; exp(-k)OPEN: Explicit? [AKS], Determine m (k2</li><li><p>Counting classes: PP, #P, P#P, C = C(Z1,Z2,,Zn) is a small circuit/formula, k=2n, </p><p> C(000) C(001) C(111) [Gill] PP[Valiant] #P C(000) C(001) C(111) </p></li><li><p>The richness of #P-complete problems</p><p>C(000) C(001) C(111) NP#P C(000) C(001) C(111) SATCLIQUE</p><p>#SAT#CLIQUEPermanent#2-SATNetwork ReliabilityMonomer-DimerIsing, Potts, TutteEnumeration, Algebra, Probability, Stat. Physics </p></li><li><p>The power of counting: Todas Theorem PHP NP PSPACE P#P </p><p>[Valiant-Vazirani] Poly-time reduction:C D</p><p>OPEN: DeterministicValiant-Vazirani?</p><p>PROBABILISTIC</p></li><li><p>Nice properties of PermanentPer is downwards self-reduciblePern(X) = Sn i[n] Xi(i) Pern(X) = i[n] Pern-1(X1i)</p><p>Per is random self-reducible[Beaver-Feigenbaum, Lipton]FnxnC errsx+3yx+2yxx+yC errs on 1/(8n)Interpolate Pern(X)from C(X+iY) with Y random, i=1,2,,n+1</p></li><li><p>Hardness amplificationIf the Permanent can be efficiently computed for most inputs, then it can for all inputs !</p><p>If the Permanent is hard in the worst-case, then it is also hard on average</p><p>Worst-case Average case reduction</p><p>Works for any low degree polynomial.Arithmetization: Boolean functionspolynomials</p></li><li><p>Avalanche of consequencesto probabilistic proof systemsUsing both RSR and DSR of Permanent!</p><p>[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</p></li><li><p>Which classes have complete RSR problems?EXPPSPACE Low degree extensions#P PermenentPHNP No Black-Box reductionsP [Fortnow-Feigenbaum,Bogdanov-Trevisan] NC2 DeterminantLNC1 [Barrington]</p><p>OPEN: Non Black-Box reductions??</p></li><li><p>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).</p><p> =1 #P = BPP =1-1/n #P = BPP [Lipton] =1/nc #P = BPP [CaiPavanSivakumar] =n3/p #P = PH =AM [FeigeLund] =1/p possible!</p><p>OPEN: Tighten the bounds!(Improve Reed-Solomon list decoding [Sudan,])</p></li><li><p>Hardness vs. Randomness[Babai-Fortnaow-Nisan-Wigderson]EXP P/poly BPP SUBEXP</p><p>[Impagliazzo-Wigderson]EXP BPP BPP SUBEXP</p><p>[Kabanets-Impagliazzo] Permanent is easy iff Identity Testing can be derandomized</p><p>Proof:EXP P/poly Were doneEXP P/poly Per is EXP-complete[Karp-Lipton,Toda]workRSRDSRwork</p></li><li><p>[Vinodchandran]: PP SIZE(n10) [Aaronson]: This result doesnt relativize[Santhanam]: MA/1 SIZE(n10)OPEN: Prove NP SIZE(n10) [Aaronson-Wigderson] requires non-algebrizing proofsVinodchandrans Proof:PP P/poly Were donePP P/poly P#P = MA [LFKN] P#P = PP 2P PP [Toda] PP SIZE(n10) [Kannan]Non-relativizing &amp; Non-naturalcircuit lower bounds</p></li><li><p>PMP(G) Perfect Matching polynomial of G[ShamirSnir,TiwariTompa]: msize(PMP(Kn,n)) &gt; exp(n)[FisherKasteleynTemperly]:size(PMP(Gridn,n)) = poly(n)[Valiant]: msize(PMP(Gridn,n)) &gt; exp(n)</p><p>The power of negation Arithmetic circuitsBoolean circuitsPM Perfect Matching function[Edmonds]: size(PM) = poly(n)[Razborov]: msize(PM) &gt; nlogn OPEN: tight?[RazWigderson]: mFsize(PM) &gt; exp(n)</p></li><li><p>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]: evaluatingsize n degree d arithmetic circuits DetOPEN: Improve to Detpoly(n,d)</p><p>The power of Determinant(and linear algebra)nlogd </p></li><li><p>Algebraic analog of PNPF field, char(F)2.XMk(F) Detk(X) = Sk sgn() i[k] Xi(i)YMn(F) Pern(Y) = Sn i[n] Yi(i)</p><p>Affine map L: Mn(F) Mk(F) is good if Pern = Detk Lk(n): the smallest k for which there is a good map?</p><p>[Polya] k(2) =2 Per2 = Det2 </p><p>[Valiant] F k(n) &lt; exp(n)[Mignon-Ressayre] F k(n) &gt; n2[Valiant] k(n) poly(n) PNP[Mulmuley-Sohoni] Algebraic-geometric approach</p></li><li><p>Detn vs. Pern</p><p>[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 </p><p>[Ryser] Pern has depth-3 circuits of size n22nOPEN: Improve n! for Detn</p></li><li><p>Approximating Pern</p><p>A: nn 0/1 matrix. B: Bij Aij at random[Godsil-Gutman] Pern(A) = E[Detn(B)2][KarmarkarKarpLiptonLovaszLuby] variance = 2nB: Bij AijRij with random Rij, E[R]=0, E[R2]=1Use R={,2,3=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)</p></li><li><p>Approx Pern deterministicallyA: 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 DSOPEN: Find a deterministic subexp approx.</p></li><li><p>Many happy returns, Les !!!</p><p>***********Problem is NP complete, but this is no lower bound**Last two use Reed-Solomon decoding.*The Permanent (due to its properties) played a crucial role in the proofs of many theorems.Here, Hardness vs Randomness under uniform assumptions*And here, for circuit lower bounds.****Often our techniques cannot distinguish Determinant from permanentand in some sense they even look equal in Natures eyes.In one model depth 3 arithmetic circuits, permanent seems easier!</p><p>*The previous lecture (by Jerrum) covered approximations via Monte Carlo Markov Chains **</p></li></ul>