Subhash Khot Georgia Tech

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SDP Gaps and UGC-Hardness for Max-Cut-Gain. Subhash Khot Georgia Tech. Ryan O’Donnell Carnegie Mellon. &amp;. Max-Cut: Weighted graph H (say weights sum to 1). Find a subset of vertices A to maximize weight of edges between A and A c. A. .097. .183. .059. [Trivial algorithm]

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SDP Gaps and

UGC-Hardness

for Max-Cut-Gain

Subhash KhotGeorgia Tech

Ryan O’DonnellCarnegie Mellon

&

Max-Cut:
• Weighted graph H(say weights sum to 1).
• Find a subset of vertices Ato maximize weight ofedges between A and Ac.

A

.097

.183

.059

[Trivial algorithm]

[Karp’72]:5/6vs.5/6 − 1/poly(n)NP-hard

[Sahni-Gonzalez’76]

[Goemans-Williamson’95]: .878 factor

[Zwick’99/FL’01/CW’04]: 1/2 + (/log(1/))

[KKMO+MOO’05]: UGC-hardness

s

• When OPT is c, can you in poly-time cut s?

1

arccos(1−2c)/

.878 c

Max-Cut-Gain

c

1/2

1

.845

s

• When OPT is c, can you in poly-time cut s?

Theorem 1:

SDP integrality gap in blue.

Theorem 2:

UGC-hardness there too.

1/2 + (11/13)

Theorem 3:

Other stuff.

1/2 + (2/)

Theorem 4:

1/2 + O(/log(1/))

1/2 + (/log(1/))

c

1/2

1/2 + 

Theme of the paper:
• Semidefinite programming integrality gaps arise naturally in Gaussian space.
• Can be translated into Long Code tests; ) UGC-hardness.
Semidefinite programming gaps
• Weighted graph: H = (V, w:V£V ! R¸0)
• Assignments: A:V ! [−1,1] vs. A:V ! Bn
• Compare:
• Goemans-Williamson: “For all H, s¸ blah(c).”
• Proof: Given A, construct A via:

(unit n-dim. ball)

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

(x,y) Ã w

A

1. Pick G, rand. n-dim. Gaussian

2. Define A(x) = sgn(G ¢ A(x))

Semidefinite programming gaps
• Weighted graph: H = (V, w:V£V ! R¸0)
• Assignments: A:V ! [−1,1] vs. A:V ! Bn
• Compare:
• Feige-Langberg/Charikar-Wirth: “For all H, s¸ blah(c).”
• Proof: Given A, construct A via:

(unit n-dim. ball)

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

(x,y) Ã w

A

1. Pick G, rand. n-dim. Gaussian

F

1

2. Define A(x) = sgn(G ¢ A(x))

2. Define A(x) = F(G ¢ A(x))

−1

Semidefinite programming gaps
• Weighted graph: H = (V, w:V£V ! R¸0)
• Assignments: A:V ! [−1,1] vs. A:V ! Bn
• Compare:
• Goemans-Williamson: “For all H, s¸ blah(c).”
• Proof: Given A, construct A via:

(unit n-dim. ball)

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

(x,y) Ã w

A

1. Pick G, rand. n-dim. Gaussian

2. Define A(x) = sgn(G ¢ A(x))

Semidefinite programming gaps
• Weighted graph: H = (V, w:V£V ! R¸0)
• Assignments: A:V ! [−1,1] vs. A:V ! Bn
• Compare:

(unit n-dim. ball)

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

(x,y) Ã w

A

Goemans-Williamson: “For all H, s¸ blah(c).”

Proof: Given A, construct A via:

1. Pick G, rand. n-dim. Gaussian

2. Define A(x) = sgn(G ¢ A(x))

Semidefinite programming gaps
• Weighted graph: H = (V, w:V£V ! R¸0)
• Assignments: A:V ! [−1,1] vs. A:V ! Bn
• Compare:

(unit n-dim. ball)

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

(x,y) Ã w

A

Feige-Schechtman: “There exists H s.t. s· blah(c).”

(matches GW for c¸ .845)

Proof: Take V = Rn, w = picking (1−2c)-correlated Gaussians.

Take A(x) = x / || x ||.

Best A is A(x) = sgn(G ¢ x), for any G.

Proof: Symmetrization. [Borell’85]

Semidefinite programming gaps
• Weighted graph: H = (V, w:V£V ! R¸0)
• Assignments: A:V ! [−1,1] vs. A:V ! Bn
• Compare:

(unit n-dim. ball)

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

(x,y) Ã w

A

This paper: “There exists H s.t. s· blah(c).”

(essentially matches FL/CW for c = 1/2 + )

Proof: Take V = Rn, w = picking mixture of 2 corr’d Gaussian pairs.

Proof: Take V = Rn, w = picking (1−2c)-correlated Gaussians.

Take A(x) = x / || x ||.

Best A is A(x) = F(G ¢ x), for any G.

Best A is A(x) = sgn(G ¢ x), for any G.

Proof: Symmetrization. [Borell’85]

Proof:

Semidefinite programming gaps
• Weighted graph: H = (V, w:V£V ! R¸0)
• Assignments: A:V ! [−1,1] vs. A:V ! Bn
• Compare:

(unit n-dim. ball)

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

(x,y) Ã w

A

Feige-Schechtman: “There exists H s.t. s· blah(c).”

(matches GW for c¸ .845)

Proof: Take V = Rn, w = picking (1−2c)-correlated Gaussians.

Take A(x) = x / || x ||.

Best A is A(x) = sgn(G ¢ x), for any G.

Proof: Symmetrization. [Borell’85]

Long code (“Dictator”) Tests

Weighted graph: H = ({−1,1}n, w:V£V ! R¸0)

Weighted graph: H = (V, w:V£V ! R¸0)

Assignments: A:{−1,1}n! [−1,1] vs. Ai(x) = xi

Assignments: A:V ! [−1,1] vs. A:V ! Bn

(unit n-dim. ball)

far from all Dictators

Compare:

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

far from all Dictators

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

i

i

i

(x,y) Ã w

A

Feige-Schechtman: “There exists H s.t. s· blah(c).”

(matches GW for c¸ .845)

Proof: Take V = Rn, w = picking (1−2c)-correlated Gaussians.

Take A(x) = x / || x ||.

Best A is A(x) = sgn(G ¢ x), for any G.

Proof: Symmetrization. [Borell’85]

Long code (“Dictator”) Tests

Weighted graph: H = ({−1,1}n, w:V£V ! R¸0)

Assignments: A:{−1,1}n! [−1,1] vs. Ai(x) = xi

far from all Dictators

Compare:

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

far from all Dictators

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

i

i

i

(x,y) Ã w

KKMO/MOO: “There exists w s.t. s· blah(c).”

Feige-Schechtman: “There exists H s.t. s· blah(c).”

(matches GW for c¸ .845)

Proof: w = picking (1−2c)-correlated bit-strings.

Proof: Take V = Rn, w = picking (1−2c)-correlated Gaussians.

Take A(x) = x / || x ||.

Best A is A(x) = sgn(G ¢ x), for almost any G.

Best A is A(x) = sgn(G ¢ x), for any G.

Proof: Somewhat elaborate reduction to [Borell’85] (“Majority Is Stablest”)

Proof: Symmetrization. [Borell’85]

Long code (“Dictator”) Tests

Weighted graph: H = ({−1,1}n, w:V£V ! R¸0)

Assignments: A:{−1,1}n! [−1,1] vs. Ai(x) = xi

far from all Dictators

Compare:

s:= max E [ (½) − (½) A(x) ¢ A(y) ]

A

(x,y) Ã w

far from all Dictators

c:= max E [ (½) − (½) A(x) ¢ A(y) ]

vs.

i

i

i

(x,y) Ã w

KKMO/MOO: “There exists w s.t. s· blah(c).”

This paper: “There exists w s.t. s· blah(c).”

(essentially matches FL/CW for c = 1/2 + )

(matches GW for c¸ .845)

Proof: w = picking mixture of 2 corr’d bit-string pairs.

Proof: w = picking (1−2c)-correlated bit strings.

Best A is A(x) = F(G ¢ x), for almost any G.

Best A is A(x) = sgn(G ¢ x), for almost any G.

Proof: Somewhat elaborate reduction to [Borell’85] (“Majority Is Stablest”)

Proof: if |ai| is small for each i.

Conclusion:
• There is something fishy going on.
• What is the connection between SDP integrality gaps and Long Code tests?