Ryan ’Donnell Carnegie Mellon University O

Ryan ’Donnell Carnegie Mellon University

Part 1: Inverse Theorems Part 2: Inapproximability Part 3: The connection

Inverse Theorem for Linearity

Fourier Analysis

Inverse Theorem for Linearity

“High-end inverse theorem”: Pr [.. ] ≥ 1− ⇒ f is (1−2)-correlated with some χ ξ “Low-end inverse theorem”: Pr [.. ] ≥ + ⇒ f is 2-correlated with some χ ξ

X Y Z = uniform on ,,,

X Y Z ,,, [Håstad’97] = draw from w.p. 1−δ unif. on all 8 w.p. δ

[Håstad’97] |ξ| = # nonzero coords in ξ e.g.: ξ = (1,0,1,0,0,…,0,1), 〈 ξ, x 〉 = x 1 +x 3 +x n, |ξ| = 3

Håstad’s low-end inverse theorem: Pr [.. ] ≥ + η ⇒ f is 2η-correlated with some sparse χ ξ

Inverse Thm: If f has o(1) correlation w/ every O(1)-sparse χ ξ [Håstad’97] then p < + o(1). (besides ξ ≠ 0) “f is quasirandom”

Inverse Thm: If f has o(1) correlation w/ every O(1)-sparse χ ξ [Håstad’97] then p < + o(1). -Verse Thm: If f = χ ξ with |ξ| = 1 then p ≥ 1 − o(1).

Problem: 3-Sat Input: I = Alg’s goal: an assignment satisfying as many constraints as possible. 3-OR

Algorithm must be “ efficient ” # steps ≤ n O(1)

For input I, Opt( I ) = fraction of constraints satisfied by best asgnmt and with algorithm “Alg”, Alg( I ) = fraction of constraints satisfied by Alg’s asgnmt

Fact:There is no efficient algorithm for 3-OR with the following guarantee: if Opt( I ) = 1 then Alg( I ) = 1. * * unless P = NP.

Q: Can we have an efficient 3-OR alg. s.t. if Opt( I ) = 1 then Alg( I ) ≥ ? A: No. * The “PCP Theorem.” [Arora-Safra’92, Arora-Lund-Motwani-Sudan-Szegedy’92]

Q: Can we have an efficient 3-OR alg. s.t. if Opt( I ) = 1 then Alg( I ) ≥ ? A: No. * “Håstad’s 3-OR Inapproximability.” [Håstad’97]

Q: Can we have an efficient 3-OR alg. s.t. if Opt( I ) = 1 then Alg( I ) ≥ ? A: Yes we can. Choose a random asgnmt. [Johnson’74]

Problem: 3-XOR Input: I = overdetermined(?) linear sys. over with 3 vbls/eqn. 3-Lin (mod 2)

Q: Can we have an efficient 3-XOR alg. s.t. if Opt( I ) = 1 then Alg( I ) = 1 ? A: Yes. Gaussian Elimination.

Håstad’s 3-XOR Inapproximability Theorem: There is no * efficient 3-XOR alg. s.t. if Opt( I ) ≥ 1−δ then Alg( I ) ≥ +η. Remark: There is an efficient alg. with Alg( I ) ≥ always. Pick either x ≡ 0 or x ≡ 1.

Max-Cut

Problem: Max-Cut Input: I = (“2-≠”)

The Goemans-Williamson Algorithm: [GW’94] There is an efficient Max-Cut alg. s.t. ∀ ρ ≥.844, if Opt( I ) = ρ then Alg( I ) ≥ 1 1 ½.844

Max-Cut Inapproximability Theorem: There is no * * better efficient algorithm. [Khot-Kindler-Mossel-O’04, Mossel-O-Oleszkiewicz’05]

Inverse Thm: [Håstad’97] If f = χ ξ with |ξ| = 1 then p ≥ 1 − o(1). then p < + o(1). If f is quasirandom -Verse Thm:

then p < + o(1). If f is quasirandom If f = χ ξ with |ξ| = 1 then p ≥ 1 − o(1). Inverse Thm. Inapprox. There is no * efficient 3-XOR alg. s.t. if Opt( I ) ≥ 1 − o(1) then Alg( I ) ≥ + o(1).

then p < + o(1). If f is quasirandom If f = χ ξ with |ξ| = 1 then p = 1. Inverse Thm.

then p < + o(1). If f is quasirandom If f = χ ξ with |ξ| = 1 then p = 1. Inverse Thm. Inapprox. There is no * efficient 3-OR alg. s.t. if Opt( I ) = 1 then Alg( I ) ≥ + o(1).

then p < + o(1). If f is quasirandom If f = χ ξ with |ξ| = 1 then p = 1. Inverse Thm. Inapprox. There is no * efficient 3-OR alg. s.t. if Opt( I ) = 1 then Alg( I ) ≥ + o(1).

then p < + o(1). If f is quasirandom * If f = χ ξ with |ξ| = 1 then p = ρ. Inverse Thm. (sharp: f = Majority) “Majority Is Stablest” [Mossel-O-Oleszkiewicz’05]

then p < + o(1). If f is quasirandom * If f = χ ξ with |ξ| = 1 then p = ρ. Inverse Thm. Inapprox. There is no * * efficient Max-Cut (i.e., “ 2-≠ ”) alg. s.t. if Opt( I ) = ρ then Alg( I ) ≥ + o(1).

then p < + o(1). If f is quasirandom * If f = χ ξ with |ξ| = 1 then p = ρ. Inverse Thm. Inapprox. There is no * * efficient Max-Cut (i.e., “ 2-≠ ”) alg. s.t. if Opt( I ) = ρ then Alg( I ) ≥ + o(1). [one-semester course]

Ask me about… Invariance Principle [MOO’05, Mossel’08] (“CLT for quasirandom * polynomials”) Geometry of Gaussian Space [Borell’85] Unique Games Conjecture * * [Khot’02] Connections to Voting / Social Choice ( Influences [Banzhaf’65], Arrow’s Theorem [Kalai’02], Ain’t Over Till It’s Over Theorem [MOO’05] ) New inverse theorem & inapproximability for the 3-Any problem, 1 vs. ⅝ [O-Wu’09]