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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
X
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D = uniform on
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,, ,
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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]
