Zwick

1
Zwicks Conjectureis implied by most of Khots
Conjectures
(aka, Conditional hardness for satisfiable
3-CSPs)
Ryan ODonnell Yi Wu Carnegie Mellon University
2

• Zwicks Conjecture 1997
• NP ? naPCP1,5/8?(O(log n), 3).
• Every language in NP has a probabilistically
  checkable proof system of polynomial size in
  which the verifier queries 3 bits of the proof
  nonadaptively, accepts correct proofs with
  probability 1, and accepts incorrect proofs with
  probability at most 5/8?.

For all ? gt 0,
3

• Zwicks Conjecture 1997
• NP ? naPCP1,5/8?(O(log n), 3).
• Given a satisfiable 3CSP, its NP-hard to
  satisfy 5/8? of the constraints.

4

• 3CSPs

5

• Zwicks Conjecture 1997
• NP ? naPCP1,5/8?(O(log n), 3).
• 3 minimal.
• 1 natural, for proof systems.
• na natural, for CSP
  inapproximability.
• 5/8 this is the conjecture.

6

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
0
1
1/8
Cook71
Johnson73
7

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
1/8
.999999
AS92,ALMSS92
Johnson73
8

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
.299
.8999
BGS95
BGS95
9

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
.367
.8999
TSSW96
BGS95
10

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
.367
3/4?
Håstad97
TSSW96
11

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
.514
3/4?
Håstad97
Trevisan97
12

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
5/8
3/4?
Håstad97
Zwick97
13

• Zwicks Conjecture 1997
• NP ? naPCP1,5/8?(O(log n), 3).
• Given a satisfiable 3CSP, its NP-hard to
  satisfy 5/8? of the constraints.

14

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
5/8
3/4?
Håstad97
Zwick97
15

• Approximating Satisfiable 3CSPs

NP-hard
In BPP
5/8
20/27?
KS06
Zwick97
16

• Approximating Satisfiable 3CSPs

NP-hard
Assuming any Khot D-to-1 Conjecture
In BPP
5/8
20/27?
KS06
Zwick97
OW09
17

• Remind me of Khots D-to-1 Conjectures?

18

• Label-Cover
• p1(V1) U4p2(V3) U2p3(V3) U9p4(V2) U4

Ui vbls over m
Input
Vi vbls over Dm
pj maps are D-to-1
Razs Theorem ? d gt 0, if m
poly(1/d) and D poly(1/d), then NP-hard to
tell satble from d-satble.
19

• Label-Cover
• p1(V1) U4p2(V3) U2p3(V3) U9p4(V2) U4

Ui vbls over m
Input
Vi vbls over Dm
pj maps are D-to-1
2-to-1 Conjecture Khot02 ? d gt 0,
if m poly(1/d) and D 2, then NP-hard
to tell satble from d-satble.
20

• Label-Cover
• p1(V1) U4p2(V3) U2p3(V3) U9p4(V2) U4

Ui vbls over m
Input
Vi vbls over Dm
pj maps are D-to-1
3-to-1 Conjecture Khot02 ? d gt 0,
if m poly(1/d) and D 3, then NP-hard
to tell satble from d-satble.
21

• Label-Cover
• p1(V1) U4p2(V3) U2p3(V3) U9p4(V2) U4

Ui vbls over m
Input
Vi vbls over Dm
pj maps are D-to-1
100-to-1 Conjecture Khot02 ? d
gt 0, if m poly(1/d) and D 100,
then NP-hard to tell satble from d-satble.
22

• Label-Cover
• p1(V1) U4p2(V3) U2p3(V3) U9p4(V2) U4

Ui vbls over m
Input
Vi vbls over Dm
pj maps are D-to-1
Unique Games Conjecture Khot02 ? d gt 0,
if m poly(1/d) and D 1, NP-hard to tell
(1-d)-satble from d-satble.
23

• Conjectures
• 2-to-1 ? 3-to-1 ? 4-to-1 ?
• ? 100-to-1 ?
• ? poly(1/d)-to-1 (true)
• None known comparable with UGC.

24

• Why not use Raghavendra08?
• UGC-based, hence cant address theory of
  satisfiable instances
• Shows Alg/UGC-hard match at some number
  but what is the number?

25

• Why not use Raghavendra08?
• Shows Alg/UGC-hard match at some number
  but what is the number?
• UGC-based, hence cant address theory of
  satisfiable instances

26

• Challenges
• Shows Alg/UGC-hard match at some number
  but what is the number?
• UGC-based, hence cant address theory of
  satisfiable instances

? Design a 1 vs. 5/8 Dictator Test for D-to-1
? Overcome perfect correlation, pairwise
dependence in Invariance Principle arguments
27

• Take-home message

If you can design a Dictator Test that
seems to work you
can make it work.
28

• Challenges

Design a 1 vs. 5/8 Dictator Test for D-to-1
Overcome perfect correlation, pairwise
dependence in Invariance Principle arguments
29

• Challenges

Design a 1 vs. 5/8 Dictator Test for D-to-1
Overcome perfect correlation, pairwise
dependence in Invariance Principle arguments
30

• Designing a 1 vs. 5/8 Dictator Test
• Q Why is Zwicks 3CSP alg. stuck at 5/8?
• A The Not-Two predicate

Linear/random 5/8 SDP alg 5/8
31

• Designing a 1 vs. 5/8 Dictator Test
• Q (Håstad01) 1 vs. 5/8? hardness for NTW?
• The Not-Two predicate

32

• Designing a 1 vs. 5/8 Dictator Test
• Q (Håstad01) 1 vs. 5/8? hardness for NTW?
• A (Our main thm.) Yes, assuming D-to-1 Conj.
• for any const. D lt 8.

33

• Designing a 1 vs. 5/8 Dictator Test
• Hås97 gave a 1-? vs. 1/2? Dictator Test
  for D-to-1, D arbitrary, using XOR.
• We give a 1 vs. 5/8? Dictator
  Test for D-to-1, D constant, using NTW.

34

• D-to-1 Dictator Tests using F
• Somehow pick corrd strings x?0,1m,
  y,z?0,1Dm.
• Test whether F( f(x), g(y), g(z) ).

m
f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
0
D 3
35

• D-to-1 Dictator Tests using F
• Completeness
• If f is ith Dictator, g is jth Dictator,
  j matches i, then Prx,y,z F( f(x),
  g(y), g(z) ) c.
• Soundness
• If f and g odd, T1-? f and T1-? g have no
  matching influential variables in common,
  then Prx,y,z F( f(x), g(y), g(z) ) s?.

36

• Completeness
• If f is ith Dictator, g is jth Dictator,
  j matches i, then Prx,y,z F( f(x),
  g(y), g(z) ) c.
• Soundness
• If f and g odd, T1-? f and T1-? g have no
  matching influential variables in common,
  then Prx,y,z F( f(x), g(y), g(z) ) s?.

Håstad F XOR, c 1-?, s 1/2
37

• Completeness
• If f is ith Dictator, g is jth Dictator,
  j matches i, then Prx,y,z F( f(x),
  g(y), g(z) ) c.
• Soundness
• If f and g odd, T1-? f and T1-? g have no
  matching influential variables in common,
  then Prx,y,z F( f(x), g(y), g(z) ) s?.

Us F NTW, c 1, s 5/8
38

• Håstads Dictator Test using XOR
• Somehow pick corrd strings x?0,1m,
  y,z?0,1Dm.
• Test whether XOR( f(x), g(y), g(z) ).

m
f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
0
D 3
39

• Håstads Dictator Test using XOR
• Pick blocks from some dist. on 0,1 x 0,1D x
  0,1D.
• Test whether XOR( f(x), g(y), g(z) ).

m
f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
0
D 3
40

• Håstads Dictator Test using XOR
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Test whether XOR( f(x), g(y), g(z) ).

m
f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
0
D 3
41

• Håstads Dictator Test using XOR
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak Rerandomize each zi with prob. 2?.

m
f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
1
D 3
42

• Completeness
• If f is ith Dictator, g is jth Dictator,
  j matches i, then Prx,y,z F( f(x),
  g(y), g(z) ) c.
• Soundness
• If f and g odd, T1-? f and T1-? g have no
  matching influential variables in common,
  then Prx,y,z F( f(x), g(y), g(z) ) s?.

Håstad F XOR, c 1-?, s 1/2
43

• Completeness
• Each column (xi,yj,zj) satisfies XOR w.p.
  1-?.
• Soundness
• If f and g odd, T1-? f and T1-? g have no
  matching influential variables in common,
  then Prx,y,z F( f(x), g(y), g(z) ) s?.

Håstad F XOR, c 1-?, s 1/2
44

• Completeness
• Each column (xi,yj,zj) satisfies XOR w.p.
  1-?.
• Soundness
• Seems like it should work. If f g
  Majority, or f g Parity then Prx,y,z
  XOR( f(x), g(y), g(z) ) 1/2o(1).

Håstad F XOR, c 1-?, s 1/2
45

• Håstads Dictator Test using XOR
• At a technical level
• Håstad does direct Fourier Analysis.
• We sketch an Invariance Principle proof which
  works for D O(1).
• (Have to reprove MOO05,Mos08.)

46

• Håstads Dictator Test using XOR
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak Rerandomize each zi with prob. 2?.
• Within a block, imperfect correlation between x
  (y,z).
• So E f(x) g(y) g(z) E Tf(x) Tg(y)
  Tg(z) .

f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
1
47

• Håstads Dictator Test using XOR
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak Rerandomize each zi with prob. 2?.
• Invariance Tf, Tg have no matching influential
  vbl.
• ? can change dist. to anything w/ same 2-wise
  corrs.

f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
1
48

• Håstads Dictator Test using XOR
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Then rerandomize x1.
• Tweak Rerandomize each zi with prob. 2?.
• E Tf(x) Tg(y) Tg(z) E Tf(x) E Tg(y)
  Tg(z) 0.

f (
x
1
1
0
1
1
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
1
49

• Completeness
• Each column (xi,yj,zj) satisfies XOR w.p.
  1-?.
• Soundness
• Seems like it should work. If f g
  Majority, or f g Parity then E f(x)
  g(y) g(z) 0.

Håstad F XOR, c 1-?, s 1/2
50

• Completeness
• Each column (xi,yj,zj) satisfies NTW w.p. 1.
• Soundness
• Seems like it should work. If f g
  Majority, or f g Parity then E f(x)
  g(y) g(z) 0.

Us F NTW, c 1, s 5/8
51

• Håstads Dictator Test using XOR
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• For NTW completeness, okay if column is (0,0,0).

f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
0
52

• Our Dictator Test using NTW
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak W.p. ?, make a random column all x1.

f (
x
1
0
1
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
1
0
1
0
0
53

• Our Dictator Test using NTW
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak W.p. ?, make a random column all
  x1.
• Unfortunately perfect correlation between x
  (y,z).
• (Even for D 2. Imperfect for D 1, hence
  OW09a.)

f (
x
1
0
1
1
0
g (
y
1
0
1
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
1
0
0
0
0
1
0
1
0
0
1
0
0
54

• Our Dictator Test using NTW
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak W.p. ?, make a random column all
  x1.
• Escape hatch imperfect correlation between
  (x,y) z.
• So E f(x) g(y) g(z) E f(x) g(y) Tg(z)
  .

f (
x
1
0
1
1
0
g (
y
1
0
1
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
1
0
0
0
0
1
0
1
0
0
1
0
0
55

• Our Dictator Test using NTW
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak W.p. ?, make a random column all
  x1.
• Escape hatch imperfect correlation between
  (x,y) z.
• So E f(x) g(y) g(z) E f(x) Tg(y) Tg(z)
  .

f (
x
1
0
1
1
0
g (
y
1
0
1
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
1
0
0
0
0
1
0
1
0
0
1
0
0
56

• Our Dictator Test using NTW
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak W.p. ?, make a random column all
  x1.
• So E f(x) g(y) g(z) E f(x) Tg(y) Tg(z)
  .
• Now noise on (y,z) ? as if imperfect corr. for
  x (y,z).

f (
x
1
0
1
1
0
g (
y
1
0
1
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
1
0
0
0
0
1
0
1
0
0
1
0
0
57

• Our Dictator Test using NTW
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak W.p. ?, make a random column all
  x1.
• So E f(x) g(y) g(z) E f(x) Tg(y) Tg(z)
  .
• E Tf(x)
  Tg(y) Tg(z) .

f (
x
1
0
1
1
0
g (
y
1
0
1
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
1
0
0
0
0
1
0
1
0
0
1
0
0
58

• Our Dictator Test using NTW
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Tweak W.p. ?, make a random column all
  x1.
• Invariance Tf, Tg have no matching
  influential vbl.
• ? can change dist. to anything w/ same 2-wise
  corrs.

f (
x
1
0
1
1
0
g (
y
1
0
1
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
1
0
0
0
0
1
0
1
0
0
1
0
0
59

• Our Dictator Test using NTW
• Blocks x1, y1,y2,,yD unif. random, zi
  x1?yi?1.
• Then rerandomize x1.
• Tweak W.p. ?, make a random column all
  x1.
• E Tf(x) Tg(y) Tg(z) E Tf(x) E Tg(y)
  Tg(z) 0.

f (
x
0
0
0
1
0
g (
y
1
0
0
0
1
1
0
0
0
0
1
0
0
0
1
g (
z
1
0
0
0
0
0
0
1
0
1
0
0
1
0
0
60

• Take-home message

If you can design a Dictator Test that
seems to work you
can make it work.
61

• Open technical problems
• We need 1 vs. 2-2O(D2) hardness for D-to-1.
  Probably could get away with 2-poly(D). With
  1/poly(D)?
• Use D-to-1 for other problems.Max-NAE3 with
  perfect completeness?