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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