1. Jeff Kinne Indiana State University Part I: Feb 11, 2011 Part II: Feb 25, 2011

2. Note: pictures on the board… 1

3. P – Polynomial Time n: “size of input” Count number of “basic operations” Addition: O(n) Multiplication: O(n 2 ) Shortest path: O(n) 2-coloring (bipartness): O(n) Matrix multiplication: O(n 3/2 ) Determinant: O(n 3/2 ) 2

4. P – Polynomial Time Poly size circuit of AND, OR, NOT gates x1x1 x2x2 x3x3 3

5. NP – Nondeterministic Poly time Give me the answer, I can check it in poly time 3-coloring: verify in O(n) time factoring: verify in O(n 2 ) time theorem proving, bin packing, traveling salesperson, integer programming, graph isomorphism, … optimization problems ! 4

6. NP – Nondeterministic Poly time x1x1 x2x2 x3x3 c1c1 c2c2 5

7. 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 6

8. 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 7

9. 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 8

10. Algebraic-P Poly size circuit of *, + gates, field elements, poly deg + * * + + x1x1 x2x2 x3x3 5 9

11. Algebraic-P Matrix multiplication Determinant All poly-size formulas are projection of det [Valiant] 10

12. Algebraic-NP 11

13. Algebraic-NP Permanent All of Alg-NP are projections of perm [Valiant] Conjecture: perm is not the projection of m x m det for any m = 2 O(log(2n)) [Valiant] Would imply Alg-P ≠ Alg-NP 12

14. Results f(x 1, x 2, …, x n ) = x 1 r + x 2 r + … + x n r requires size Ω(n*log(r)) [Strassen] There exists f, deg r, requires size [Hrubeš, Yehudayoff] 13

15. Structural results for Alg-P All intermediate gates homogeneous polynomials [Strassen], [Raz] Remove divisions [Strassen] Depth O(log 2 (n)) [Valiant, Skyum, Berkowitz, Rackoff] 14

16. Restricted Settings Depth-3,, Mod-q requires size 2 Ω(n) [Grigoriev, Karpinski, Razborov] Multi-linear formulas permanent, determinant require size n Ω(n) [Raz] Monotone (positive coefficients) permanent requires size 2 Ω(n) [Jerrum, Snir] 15

17. Part II: Lower Bounds and PIT 16

18. Using “hard” polynomials 17

19. Polynomial Identity Testing Is polynomial of poly-size circuit ? Non-zero polynomial, deg d, x i at random from T Pr[(x 1, x 2, …, x n ) = ] ≤ d/|T| [Schwartz, Zippel] 18

20. Circuit … ) Goal: is ? S 1, S 2, … S n each size << n, small pairwise Test Φ’…) If ’ small circuit for f [Kabanets, Impagliazzo] 19

21. S 1, S 2, … S n each size << n, small pairwise Φ’…) … (hybrid argument) … = …, …,x n ) – x i+1 divides factor to get circuit for f 20

22. PIT algorithm => lower bounds 21

23. If PIT in P, Perm in Alg-P… P perm in NP Perm(A) = Σ j A ij * Perm(A ij * ) Guess circuit for Perm, verify with PIT P perm is hard for size n k NEXP hard for poly size [Kabanets, Impagliazzo] [Kinne et al.] 22

24. Fin Thank you! Slides online at: http://www.kinnejeff.com/ http://www.kinnejeff.com/ Excellent survey by Amir Shpilka and Amir Yehudayoff “Arithmetic Circuits: a survey of recent results and open questions” 23