3-Query Dictator Testing

Ryan O’Donnell

Carnegie Mellon Universityjoint work with

Yi WuCarnegie Mellon University

Motivation: Max-3CSP

Constraint Satisfaction Problems (CSPs)

Input:

¢ ¢ ¢

Output: Assignment: vi 2 {0,1}

Desideratum: Satisfy as much as possible.w1

w2

w3

w4

w5

w6

w7

w8

w9

¢¢¢+

= 1

Definition: 0 · OPT · 1 is max. possible

Definition: · k vbls per constraint:

= “Max-kCSP”

Fixing “type” of constraints special cases:

Max-3Sat

Max-3Lin¢ ¢ ¢

¢ ¢ ¢

Other CSPs (essentially)

Max-3CSP

Input:

¢ ¢ ¢

Output: Assignment: vi 2 {0,1}

Desideratum: Satisfy as much as possible.w1

w2

w3

w4

w5

w6

w7

w8

w9

¢¢¢+

= 1

Definition: 0 · OPT · 1 is max. possible

Definition: · 3 vbls per constraint:

= “Max-3CSP”

Max-Blah is c vs. s easy: satisfying ¸ s when OPT ¸ c is in poly time.

Max-Blah is c vs. s hard: satisfying ¸ s when OPT ¸ c is NP-hard.

Computational Complexity of CSPs

Approximability of Max-3CSP

1

s

c01

(OPT)

[Cook71]

= NP-hard

[Johnson74]

1/8

= in poly time

[AS, ALMSS92]

[BGS95]

(.96)

[Trevisan96]

1/4

[TSSW96]

(.367)

[Håstad97]

3/4

[Trevisan97]

(.514)

[Zwick98,02]

1/2

5/8

[KS06]

(.74)

[Zwick98], on his 1 vs. 5/8 easiness result for Max-3CSP:

“We conjecture that this result is optimal.”

“… the hardest satisfiable instances of Max-3CSP [for the algorithm]

turn out to be instances in which all clauses are NTW clauses.”

[Håstad97], p. 65, Concluding remarks:

The technique of using Fourier transforms to analyze [Dictator Tests] seems very strong.

It does not, however, seem universal even limited to CSPs. In particular, an open

question that remains is to decide whether the NTW predicate is non-approximable

beyond the random assignment threshold [5/8] on satisfiable instances.

Open Problems

NTW(a,b,c) = 1 ,

# 1’s among a,b,c is zero, one, or three –

i.e., Not Two

”

“

Dictator Testing(AKA Long Code testing)

• Property Testing problem

• Query access to unknown Boolean function f : {0,1}n {0,1}

• Want to test if f is a Dictator:

f(x1, …, xn) = xi for some i.

• Can only make a constant number of queries

• And by constant, I mean 3

• Or fewer

• And the queries must be non-adaptive

Dictator Testing [BGS95]

3-Query Dictator Testing

randomly chooses:

i) 3 strings, x, y, z 2 {0,1}n,

ii) a 3-bit predicate, φ :{0,1}3 → {acc, rej}

x, y, z

f(x), f(y), f(z)

“accepts” iff φ(f(x), f(y), f(z)) = acc

“Completeness” ¸ c $ all n Dictators accepted w. prob. ¸ c

“Soundness” · s $ “very non-Dictatorial f” accepted “w. prob. · s + o(1)”

Tester

“Tester uses predicate set Φ” $ Φ = {possible φ’s tester may choose}

Soundness Condition

Usually: “Every f which is ±-far from all Dictators is accepted w. prob. · s.”

[Håstad97]: Too hard! Relax.

Definition: f is quasirandom if

fixing any O(1) input bits changes bias by at most o(1).

Remark: Dictators are the epitome of not being quasirandom.

Formally: f is (²,±)-quasirandom if for all 0 < |S| · 1/±.

Quasirandomness

Definition: f is quasirandom if

fixing any O(1) input bits changes bias by at most o(1).

Not quasirandom: Dictators

“Juntas”

Epitome of quasirandom: Constants (f ´ 0, f ´ 1)

Majority

Large Parities: f(x) = where |S| > ω(1)

Dictator-vs.-quasirandom Tests

“Dictator-vs.-quasirandom” Tests:

Formally: Given a sequence of tests ( Tn),

Soundness · s $ every quasirandom f accepted w. prob. · s + o(1)

Soundness · s $ for all ´ > 0, exists ², ± > 0, for all suff. large n,

Tn accepts every (²,±)-quasirandom f w. prob. · s + ´

Meta-Theorem:

Suppose you build a Dictator-vs.-quasirandom test with:

completeness ¸ c, soundness · s,

tester uses predicate set Φ.

Then Max-Φ is c vs. s + ² hard.

(Max–Φ is the CSP where all constraints are from the set Φ.)

Connection to Inapproximability

[Zwick98], on his 1 vs. 5/8 easiness result for Max-3CSP:

“We conjecture that this result is optimal.”

“… the hardest satisfiable instances of Max-3CSP [for the algorithm]

turn out to be instances in which all clauses are NTW clauses.”

[Håstad97], p. 65, Concluding remarks:

The technique of using Fourier transforms to analyze [Dictator Tests] seems very strong.

It does not, however, seem universal even limited to CSPs. In particular, an open

question that remains is to decide whether the NTW predicate is non-approximable

beyond the random assignment threshold [5/8] on satisfiable instances.

Implication for Max-3CSP

”

“

Theorem:

a. There is a 3-query Dictator-vs.-quasirandom test, using NTW predicate,

with completeness c = 1 and soundness s = 5/8. [Pf: Fourier analysis.]

b. Every 3-query Dictator-vs.-quasirandom test, using any mix of predicates,

with completeness c = 1 has soundness s ¸ 5/8. [Pf: Uses Zwick’s SDP alg.]

Not a Theorem: Max-NTW is 1 vs. 5/8 hard.

Why? Meta-Theorem problematic… maybe with Khot’s “2-to-1 Conjecture”…??

Our Results

Our NTW-based test: how and why

3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW

f (f (f (

NTW (

p q r s t

Test: Choose triple (x, y, z) from D n.

D =

w. prob.

=

) )

)) z

yx

3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW

f (f (f (

NTW (

p

Test: Choose triple (x, y, z) from D n.

D =

w. prob.

=

) )

)) z

yx

3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW

f (f (f (

NTW (

Test: Choose triple (x, y, z) from D n.

D =

w. prob.

=

) )

)) z

yx

3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW

f (f (f (

NTW (

Test: Choose triple (x, y, z) from D± n.

D =

w. prob.

=

) )

)) z

yx

±

D = ±

Fact: (1 – ±) D + ± D XOR EQU

Equivalent test: 1. Form “random restriction” fw with ¤-probability 1 – ±.

2. Do BLR test on fw, but also accept (0,0,0).

Analyzing the Test

Pr[acc. odd f] ·

Håstad’s term: · ± when f is (±2,±2)-quasirandom

Handle with careful use of the “hypercontractive inequality”

Long story short: last term always

Open Problems

• Prove Max-3CSP is 1 vs. 5/8 + ² hard.

• Prove Max-3CSP is 1 vs. 5/8 + ² hard assuming Khot’s 2-to-1 Conjecture.

• Tackle Max-2Sat. [cf. Austrin07a, Austrin07b]

• Max-4CSP?

Open Problems