Algebraic P versus NP

Lower Bounds and PITJeff Kinne

Indiana State University

Part I: Feb 11, 2011Part II: Feb 25, 2011

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Note: pictures on the board…

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P – Polynomial Timen: “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)

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P – Polynomial TimePoly size circuit of AND, OR, NOT gates

x1

x2

x3

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NP – Nondeterministic Poly timeGive me the answer, I can check it in poly time

3-coloring: verify in O(n) time

factoring: verify in O(n2) time

theorem proving, bin packing, traveling salesperson, integer programming, graph isomorphism, …

optimization problems !

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NP – Nondeterministic Poly timePoly size circuit of AND, OR, NOT gatesRegular input x, certificate cc cause circuit = 1?

x1

x2

x3

c1 c2

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P versus NP – Who Cares?Clay Math Institute Millenium Prize

($1,000,000)

If P = NP …No security/privacyPerfect optimization

If P ≠ NP …Secruity/privacy maybeSome optimization problems really hard

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P versus NP – what we knowNot 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

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Algebraic P versus NPEfficiency 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

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Algebraic-PPoly size circuit of *, + gates, field elements,

poly deg

+

* *

++

x1 x2 x3 5

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Algebraic-P

Matrix multiplication

Determinant

All poly-size formulas are projection of det [Valiant]

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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}

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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

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Results

f(x1, x2, …, xn) = x1r + x2

r + … + xnr

requires size Ω(n*log(r)) [Strassen]

There exists f, deg r, requires size [Hrubeš, Yehudayoff]

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Structural results for Alg-P

All intermediate gates homogeneous polynomials[Strassen], [Raz]

Remove divisions [Strassen]

Depth O(log2(n))[Valiant, Skyum, Berkowitz, Rackoff]

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Restricted SettingsDepth-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]

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Part II: Lower Bounds and PIT

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Using “hard” polynomials

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Polynomial Identity Testing

Is polynomial of poly-size circuit ?

Non-zero polynomial , deg d, xi at random from T Pr[(x1, x2, …, xn) = ] ≤ d/|T| [Schwartz, Zippel]

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Hard poly f PIT algorithmCircuit … ) Goal: is ?

S1, S2, … Sn each size << n, small pairwise

Test Φ’…)

If ’ small circuit for f

[Kabanets, Impagliazzo]

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’ small circuit for fS1, S2, … Sn each size << n, small

pairwise

Φ’…)… (hybrid argument) … = …, …,xn)

– xi+1 divides factor to get circuit for f

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PIT algorithm => lower bounds

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If PIT in P, Perm in Alg-P… Pperm in NP

Perm(A) = Σj Aij * 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.]

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FinThank 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”