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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-PMatrix 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-NPPermanent
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-PAll 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”