ryan o’donnell carnegie mellon university a doctoral thesis, by per austrin kth school of computer...

Ryan O’Donnell

Carnegie Mellon University

a doctoral thesis, by

Per Austrin

KTH School of Computer Science

and Communication

opponent:

Ryan O’Donnell


opponent:

?

(Gödel Prize,

Royal Swedish

Acad. of Sci.)

(Gödel Prize x 2)

(Turing Award)

Ryan O’Donnell


opponent:

?

(Gödel Prize,

Royal Swedish

Acad. of Sci.)

(Gödel Prize x 2)

(Turing Award) Theoretical Computer Science:

Theoretical Computer Science:

Which algorithmic problems

can be solved efficiently?

Problem: 3Sat

Input:

Alg’s goal: an assignment satisfying as

many constraints as possible.

“Efficient”

= “polynomial time”

= # steps always ≤ nC

Input:

Obvious algorithm: ≈ 2n steps.

Question: Doable in nC steps?

Answer: No.

Cook’s Theorem: 3Sat is “NP-hard”

NP-hard = Not doable in polynomial time

assuming “P ≠ NP”.

“P ≠ NP”: Everyone knows it’s true.

Polynomial Time

Maximum Matching

Linear Programming

Primality

·····

1000’s of problems

NP-hard

3Sat

Traveling Salesperson

Chromatic Number

·····


an

y n

atu

ral p

rob

lem

s in

here

?

Not known to be in P or NP-hard

1. Factoring

2. Graph Isomorphism

3. · · · · · ?Handbook on Algorithms and Theory of Computation [ALR99]:

“The vast majority of natural problems in NP have resolved themselves as being either in P or NP-complete. Unless you uncover a specific connection to one of [the above] intermediate problems, it is more likely offhand that your problem simply needs more work.”

NP-Completeness Column [Joh05]:

3. Precedence Constrained 3-Processor Scheduling

Exact optimization

Approximation?

NP-hard

3Sat

Traveling Salesperson

Chromatic Number

·····


Approximation?

95%-approximating 2Sat ?

90%-approximating 2CSP ?

15%-approximating 6CSP ?

Not known to be in P or to be NP-hard.Not known to be in P or to be NP-hard.

Results from Austrin’s Thesis

95%-approximating 2Sat ? Hard.

90%-approximating 2CSP ? Hard.

15%-approximating 6CSP ? Hard.


95%-approximating 2Sat ? Hard.*

90%-approximating 2CSP ? Hard.*

15%-approximating 6CSP ? Hard.*

* Not “NP-hard”, only “UG-hard”.


95%-approximating 2Sat Hard.*

94.01656724%-approximating 2Sat Hard.*

Theorem [LLZ’02]:

94.01656724%-approximating 2Sat

can be done in polynomial time.αLLZ

αLLZ +

Definition of αLLZ

= .9401656724…

This is* the approximability threshold

of efficient algorithms for 2Sat!

Remainder of the talk:

1. Definitions

2. Statements of main results

3. Remarks about proof techniques

Max-CSP(P) Constraint Satisfaction Problem

Max-CSP(P) Villkorssatisfieringsproblem

Max-CSP(P) P : {0,1}k {acc, rej}

Input

“constraints”

Max-CSP(P) P : {0,1}k {acc, rej}

Examples:

Max-kSat: P = “ORk”

Max-kLin: P = “XORk”

Max-kAND: P = “ANDk”

Max-kCSP: any mix of k-ary preds

Max-CSP(P)

Many many many natural variants exist:

- constraints have different “weights”

- negated variables not allowed

- variables are {0, 1, 2, …, q-1}-valued

- have to use values {0, …, q-1}

“frugally”

Approximation Algorithms

α-approximation algorithm:

On input I , guaranteed to output assignment

satisfying ≥ α · Opt(I ) constraints.

Goal: find poly-time such algorithms,

or, prove it’s NP-hard


Trivial approximation for Max-CSP(P):

α-approximation, where

(Because choosing x1, …, xn randomly satisfies

α-fraction of all constraints in expectation.)

E.g.: (3/4)-approximation, for Max-2Sat.


“Max-CSP(P) is approximation-resistant”:

= “Non-trivial approximation is NP-hard.”

E.g.: Max-3Sat is approximation-resistant.

[Håstad’97]

Pairwise Independence

Let μ be a probability distribution on {0,1}k.

We say μ is pairwise independent if the

marginal on (Xi, Xj) is uniform on {0,1}2,

for all 1 ≤ i < j ≤ k, when (X1, …, Xk) ~ μ.

UG-hard

A problem is said to be “UG-hard” if it is at least

as hard as the “Unique-Label-Cover Problem”.

UG Conjecture [Khot’02]:

“The Unique-Label-Cover Problem is NP-hard.”

Outstanding open problem in TCS,

b/c we don’t “know” the answer.


1. Definitions



Thesis Main Results

1. 2-CSP hardness

2. Approximation-resistant k-CSPs

3. Randomly supported pairwise independence

4. A technical result I’ll mention only briefly

2-CSP Hardness

Let P : {0,1}2 {acc, rej}.

Let β(P) = min [somewhat complicated numerical program].

Then

β(P)-approximating Max-CSP(P) is UG-hard.

positive configuration family Θ

[Result 1]

2-CSP Hardness

In particular:

• β(OR2) = αLLZ = .94016…

(matching the [LLZ’02] algorithm)

• β(AND2) ≤ .87434…,

(nearly matching the .87401…-approx.

algorithm for Max-CSP(AND2) [LLZ’02])

[Result 1]

More on 2-ary Max-CSP(P)

Let β(P) = min [somewhat complicated numerical program].

Let α(P) = min [somewhat complicated numerical program].

Theorem: ∃ poly-time α(P)-approx alg.

Conjecture: α(P) = β(P) for all 2-ary P.

Then assuming the UG Conjecture,

β(P)-approximating Max-CSP(P) is hard.

positive configuration family Θ

all configuration families Θ

[Result 1]

Presaged…

[Raghavendra’08]:

Let γ(P) = min [very complicated numerical program],

α(P) ≤ γ(P) ≤ β(P).

Theorem: ∃ poly-time γ(P)-approx alg.

and also (γ(P)+)-approximating is UG-hard.


β(P)-approximating Max-CSP(P) is hard.

[Result 1]

Approximation-resistant k-CSPs

Let P : {0,1}k {acc, rej}.

Suppose ∃ pairwise independent distribution

μ on {0,1}k such that supp(μ) ⊆ P-1(acc).


Max-CSP(P) is approximation-resistant.

[Result 2,

with Mossel]


Q: How small a subset of {0,1}k can

support a pairwise independent distribution?

A: RoundUp4(k) points suffice

(assuming the Hadamard

Conjecture).

[Result 2]


( +)-UG-hardness for some 6-ary

CSP

( +)-UG-hardness for some 7-ary

CSP

( +)-UG-hardness for some k-ary

CSP

[Result 2]

Cor’s:

Previous best: , , NP-hardness [ST’00].

Best alg.: -approx. for Max-kCSP [CMM’07].

Randomly supported pairwise independence

Q: Does a random subset of {0,1}k

of size S support a pairwise indep.

distr.?

Thm: Yes, whp, if S ≥ C · k2.

No, whp, if S ≤ c · k2.

[Result 3,

with Håstad]

More generally…

Q: Does a random subset of {0, 1, …, q-1}k

of size S support a pairwise indep.

distr.?

Thm: Yes, whp, if S ≥ C(q) · k2.

No, whp, if S ≤ c(q) · k2.

[Result 3]

(& slightly weaker results for t-wise independence)


1. Definitions



Proof remarks for Result 3

Thm: C(q) k2 random pts in {0, 1, …, q-1}k

whp support a pairwise indep. distr.

Pf sketch: Need to show a certain random

convex body in ℝq2k2 contains origin whp.

Uses “hypercontractivity” to show that

quadratic polys of discrete rv’s are

fairly concentrated around expectation.

Proofs for Hardness Results, 1 & 2

[Håstad’97] method for showing hardness:

PCP Technology Discrete Fourier

(“Label-Cover” is NP-hard) Analysis Wizardry+


Post-2002 method for showing hardness*:

PCP Technology Discrete Fourier

(“Label-Cover” is NP-hard) Analysis Wizardry+

UG Conjecture [Khot’02]

(“Unique-L-C is NP-hard”)

“Invariance Principle”

[MOO’05,Mos’08]


Post-2002 method had led to some new results:

• .87856… UG-hardness for “Max-Cut”

• UG-hardness of C-coloring 3-colorable

graphs (for all const C)

Based on “straightforward” use of Invariance Principle.


Key to Austrin’s new hardness results:

Heroically exploit the somewhat scary

Invariance Principle to its ultimate limits.

(Thesis Result 4: Preliminary work on Invariance

Principle generalization.)

Thanks for your attention.

Time for questions?

ryan o’donnell carnegie mellon university a doctoral thesis, by per austrin kth school of computer...

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