Algebraic P versus NP Lower Bounds and PIT

Algebraic P versus NPLower Bounds and PIT Jeff Kinne Indiana State University Part I: Feb 11, 2011 Part II: Feb 25, 2011

Note: pictures on the board…

P – Polynomial Time • n: “size of input” • Count number of “basic operations” • Addition: O(n) • Multiplication: O(n2) • Shortest path: O(n) • 2-coloring (bipartness): O(n) • Matrix multiplication: O(n3/2) • Determinant: O(n3/2)

P – Polynomial Time • Poly size circuit of AND, OR, NOT gates x1 x2 x3

NP – Nondeterministic Poly time • Give me the answer, I can check it in poly time • 3-coloring: verifyin O(n) time • factoring: verifyin O(n2) time • theorem proving, bin packing, traveling salesperson, integer programming, graph isomorphism, … • optimization problems !

NP – Nondeterministic Poly time • Poly size circuit of AND, OR, NOT gates • Regular input x, certificate c • c cause circuit = 1? x1 x2 x3 c1 c2

P versus NP – Who Cares? • Clay Math Institute Millenium Prize ($1,000,000) • If P = NP … • No security/privacy • Perfect optimization • If P ≠ NP … • Secruity/privacy maybe • Some optimization problems really hard

P versus NP – what we know • Not a lot… • Results like “such and such technique is not enough” • How can we make progress? • Seek more structure, easier/simplified cases… • Algebraic P versus NP

Algebraic P versus NP • Efficiency of computing polynomials • Who cares? • If Alg-P = Alg-NP … • P=NP (and even P = BQP = PH = P#P) * caveat • If Alg-P ≠ Alg-NP … • polynomial identity testing

Algebraic-P • Poly size circuit of *, + gates, field elements, poly deg + * * + + x1 x2 x3 5

Algebraic-P • Matrix multiplication • Determinant • All poly-size formulas are projection of det[Valiant]

Algebraic-NP • ∑ in place of • Let g Algebraic-P, polynomial t • f(x1, x2, …, xn) = ∑ g(x1, x2, …, x3, w1, w2, …, wt(n)) • Sum over all possible w, each wi {0,1}

Algebraic-NP • Permanent • All of Alg-NP are projections of perm [Valiant] • Conjecture: perm is not the projection of m x m detfor any m = 2O(log(2n))[Valiant] • Would imply Alg-P ≠ Alg-NP

Results • f(x1, x2, …, xn) = x1r + x2r + … + xnr requires size Ω(n*log(r))[Strassen] • There exists f, deg r, requires size [Hrubeš, Yehudayoff]

Structural results for Alg-P • All intermediate gates homogeneous polynomials[Strassen], [Raz] • Remove divisions [Strassen] • Depth O(log2(n))[Valiant, Skyum, Berkowitz, Rackoff]

Restricted Settings • Depth-3, , Mod-q requires size 2Ω(n) [Grigoriev, Karpinski, Razborov] • Multi-linear formulas permanent, determinant require size nΩ(n) [Raz] • Monotone (positive coefficients) permanent requires size 2Ω(n) [Jerrum, Snir]

Part II: Lower Bounds and PIT

Using “hard” polynomials

Polynomial Identity Testing • Is polynomial of poly-size circuit ? • Non-zero polynomial , degd, xi at random from TPr[(x1, x2, …, xn) = ] ≤ d/|T|[Schwartz, Zippel]

Hard poly f PIT algorithm • Circuit … ) Goal: is ? • S1, S2, … Sneach size << n, small pairwise • Test Φ’…) • If ’ small circuit for f [Kabanets, Impagliazzo]

’ small circuit for f • S1, S2, … Sn each size << n, small pairwise • Φ’…) • … (hybrid argument) … • =…, …,xn) • – xi+1 divides • factor to get circuit for f

PIT algorithm => lower bounds

If PIT in P, Perm in Alg-P… • Pperm in NP • Perm(A) = ΣjAij * Perm(Aij*) • Guess circuit for Perm, verify with PIT • Pperm is hard for size nk • NEXP hard for poly size [Kabanets, Impagliazzo] [Kinne et al.]

Fin • Thank you! • Slides online at:http://www.kinnejeff.com/ • Excellent survey by Amir Shpilka and Amir Yehudayoff “Arithmetic Circuits: a survey of recent results and open questions”

Algebraic P versus NP Lower Bounds and PIT