Subhash Khot Georgia Tech

Subhash KhotGeorgia Tech

Ryan O’DonnellCarnegie Mellon

SDP Gaps and

UGC-Hardness

for Max-Cut-Gain

&

Max-Cut:

Weighted graph H

(say weights sum to 1).

Find a subset of vertices A

to maximize weight of

edges between A and Ac.

A

.059.183.0

97

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

c

s

11/2

1

[Trivial algorithm]

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

[Sahni-Gonzalez’76]

[Goemans-Williamson’95]: .878 factor

[Håstad+TSSW’97]: 17/21 vs. 16/21 NP-hard

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

[KKMO+MOO’05]: UGC-hardness

.878 c

.845

arccos(1−2c)/

Max-Cut-Gain

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

c

s

1/2 + 1/2

1/2 + (/log(1/))

1/2 + (2/)

1/2 + (11/13)

1/2 + O(/log(1/))

Theorem 1: SDP integrality gap in blue.

Theorem 2: UGC-hardness there too.

Theorem 3:

Theorem 4:Other stuff.

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)

c := max E [ (½) − (½) A(x) ¢ A(y) ]A

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.

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

2. Define A(x) = sgn(G ¢ A(x))




Compare:

Feige-Langberg/Charikar-Wirth: “For all H, s ¸ blah(c).”


(unit n-dim. ball)

c := max E [ (½) − (½) A(x) ¢ A(y) ]A

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.


2. Define A(x) = sgn(G ¢ A(x))2. Define A(x) = F (G ¢ A(x))

F1

−1




Compare:



(unit n-dim. ball)

c := max E [ (½) − (½) A(x) ¢ A(y) ]A

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.






Compare:

(unit n-dim. ball)

c := max E [ (½) − (½) A(x) ¢ A(y) ]A

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.








Compare:

(unit n-dim. ball)

c := max E [ (½) − (½) A(x) ¢ A(y) ]A

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.

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

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

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

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

(matches GW for c ¸ .845)

Proof: Symmetrization. [Borell’85]

Proof: Symmetrization. [Borell’85] Proof:




Compare:

(unit n-dim. ball)

c := max E [ (½) − (½) A(x) ¢ A(y) ]A

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.

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

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.

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




Compare:

(unit n-dim. ball)

c := max E [ (½) − (½) A(x) ¢ A(y) ]A

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.

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






(unit n-dim. ball)

c := max E [ (½) − (½) A(x) ¢ A(y) ]A

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.

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






Long code (“Dictator”) Tests



Compare:

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

Assignments: A : {−1,1}n ! [−1,1] vs. Ai(x) = xifar from all Dictators

far from all Dictators

i i i

c := max E [ (½) − (½) A(x) ¢ A(y) ]

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.

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







Compare:




i i i

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

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

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

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

c := max E [ (½) − (½) A(x) ¢ A(y) ]

s := max E [ (½) − (½) A(x) ¢ A(y) ]A (x,y) Ã w

(x,y) Ã wvs.



Compare:




i i i

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

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

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

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

This paper: “There exists w s.t. s · blah(c).” (essentially matches FL/CW for c = 1/2 + )

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

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

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

Conclusion:

There is something fishy going on.

What is the connection between SDP integrality gaps

and Long Code tests?

Subhash Khot Georgia Tech

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