Happy 60 th B’day Noga

Elementary problems encoding computational hardness Avi Wigderson IAS, Princeton or Some problems Noga never solved

Explicit object (graph, number, set,…) n = -Explainable - Reasonable - Efficiently constructible - Not random - Not generic

Linear transformations X1X1 X2X2 …. XnXn + + M: F n  F n y = Mx Y1Y1 Y2Y2 …. YnYn For most M, size(M) ≈ n 2 Challenge: Find M with size(M) ≠ O(n) c c’

Gauss complexity F any field M an n×n non-singular matrix over F. gc(M) = the smallest number of Gauss elimination steps to make M diagonal = min {s : M = E 1 E 2 … E s, E i elementary} For most M, gc(M) ≈ n 2 Challenge: Find explicit M with gc(M) ≠ O(n)

Matrix rigidity [Valiant] F any field M an n×n matrix over F. M “rigid” if a all matrices in Hamming ball around it has high rank rig(M) = the smallest number of nonzeros in a matrix S such that rank(M-S) < n/10 For most M, rig(M) ≈ n 2 Challenge: Find explicit M with rig(M) ≠ O(n)

Formula size ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ XnXn -X j XjXj -X 5 ∧ ∧ f: {0,1} n  {0,1} X1X1 X5X5 Fact: For most f, Fsize(f) ≥ exp(n) Thm Explicit f, Fsize(f) ≥ n 3 Challenge: Find f with Fsize(f) ≠ poly(n) [Andreev, Hastad,Tal]

Composition [Karchmer-Raz-W] f : {0,1} n  {0,1} g : {0,1} k  {0,1} f ° g : {0,1} nk  {0,1} Fact: For all f,g Fsize(f ° g) ≤ Fsize(f)  Fsize(g) Challenge: Prove Fsize(f ° g) ≥ α Fsize(f)  Fsize(g) for some α >0 f g gg  f g

Communication Complexity [Karchmer-W] Alice and Bob communicate to solve: Task: Find (i,j) which is an edge in one of H or G Task*: Find (i,j) which is an edge in H but not G Fact: comm(Task) ≤ O(n) bits Challenge: Prove comm(Task) ≠ O(log n) bits Thm[KW]: comm(Task) ≤ log Fsize(Hamilton) Thm[Raz-W]: comm(Task*) ≥ Ω (n) bits Alice non-Hamiltonian Hamiltonian Bob GH 0 1 1

Permanent & Determinant Det n (X) =   Sn sgn(  )  i  [n] X i  (i) Per n (X) =   Sn  i  [n] X i  (i) Easy! Hard? × × × × × × XiXi XjXj XiXi c + + f Arithmetic circuits

Determinantal complexity [Valiant] Affine map L: M n (F)  M k (F) is good if Per n = Det k  L dc(n): the smallest k for which there is a good map Thm[Polya]: dc(2) =2 Per 2 = Det 2 Thm[Valiant]: dc(n) < exp(n) Thm[Mignon-Ressayre]: dc(n) > n 2 Challenge: Prove dc(n)  poly(n) Thm[V]: Implies exponential lower bounds for Permanent a b -c d a b c d

Polynomial identities [Heintz-Schnorr,Agrawal,Kabanets-Impagliazzo] Symbolic matrix M (0,1,X 1 …,X n ) Is det(M) = 0 identically ? Small Hitting Set (of small integer substitutions). Set k=n 10 (say). H = {v 1, v 2, … v k }, a set of vectors in [k] n det(M) ≠ 0  det(M(v i )) ≠ 0 for some i Most H are small hitting sets Challenge: Find an explicit H Thm[A,KI]: Implies exponential lower bounds for Permanent X1X1 X1X1 X2X2 XnXn X5X5 XnXn X1X1 n n

Elusive curves [Raz] Everything over C (works for other fields) Fact: The moment curve avoids every hyperplane! f(x) = (x,x 2,x 3,…, x n ) of degree n, avoids deg 1 maps: for every G: C n -1  C n of degree 1, Im(f) Im(G) Challenge: Find an explicit curve of degree poly(n) which avoids all degree 2 maps. Thm[Raz]: Implies exponential lower bounds for Permanent  /

Sum-of-Squares [Hrubes-W- Yehudayoff] (X 1 2 +X … +X k 2 )(Y 1 2 +Y … +Y k 2 ) = B 1 2 +B … +B n 2 B i =B i (X,Y) Bilinear functions. n = n(k) ≤ k 2 Do better? n=1 (X 1 2 )(Y 1 2 ) = (X 1 Y 1 ) 2 n=2 (X 1 2 +X 2 2 )(Y 1 2 +Y 2 2 ) = (X 1 Y 2 +X 2 Y 1 ) 2 + (X 1 Y 1 -X 2 Y 2 ) 2 n=4 (X 1 2 +…+X 4 2 )(Y 1 2 +…+Y 4 2 ) = B 1 2 +B … +B 4 2 Euler n=8 (X 1 2 +…+X 8 2 )(Y 1 2 +…+Y 8 2 ) = B 1 2 +B … +B 8 2 Hamilton Thm[Hurwitz] 1,2,4,8 are the only ones with n=k Thm[Radon] n Z (k) < k 2 /logk Thm[James] n R (k) > 2k Thm[HWY] n Z (k) > k 6/5 Challenge: Prove n C (k) > k 1+ ε Thm[HWY]  exponential non-commutative circuit lower bound

The Fusion method Everything over GF(2) (works for other fields) M 1, M 2, … M k, a list of n×n invertible matrices S a subset of [k]. M S = ∑ i  S M i Task: cover the universe {S: S odd, M S singular} P=(P 1,P 2,P 3 ) is a 3-partition of [k]. P covers S if all three |S  P j | is odd. cov(n) = min size of covering family Challenge: Prove cov(n)  O(n) Thm[R,KW] cov(n) is a circuit lower bound for Determinant n-bit primes composite Primality [Razborov, Karchmer,W] Approximation method Finite ultrafilters Approximation method Finite ultrafilters Fractional cover O(n)

Why do Lower bounds manifest themselves in so many ways? Computation is everywhere!

Happy 60 th B’day Noga