Ryan ’Donnell Carnegie Mellon University O. Ryan ’Donnell Carnegie Mellon University.

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

Inverse Theorem for Linearity

Inverse Theorem for Linearity

Fourier Analysis

Inverse Theorem for Linearity

Inverse Theorem for Linearity

Inverse Theorem for Linearity

“High-end inverse theorem”:

Pr [ .. ] ≥ 1−ϵ ⇒ f is (1−2ϵ)-correlated

with some χ ξ

“Low-end inverse theorem”:

Pr [ .. ] ≥ + ϵ ⇒ f is 2ϵ-correlated

with some χ ξ

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= uniform on

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[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⟩ = x1+x3+xn, |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 ≤ nO(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) ≥ 0.999999 ?

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) ≥ + .000001 ?

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

1½.844

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]