subhash khot georgia tech
DESCRIPTION
SDP Gaps and UGC-Hardness for Max-Cut-Gain. Subhash Khot Georgia Tech. Ryan O’Donnell Carnegie Mellon. &. 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] - PowerPoint PPT PresentationTRANSCRIPT
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))
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)
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))2. Define A(x) = F (G ¢ A(x))
F1
−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)
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))
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)
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))
Goemans-Williamson: “For all H, s ¸ blah(c).”
Proof: Given A, construct A via:
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)
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:
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)
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).”
Proof: Take V = Rn, w = picking (1−2c)-correlated Gaussians.
(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.
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)
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]
(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]
Long code (“Dictator”) Tests
Weighted graph: H = (V, w : V£V ! R¸0)
Assignments: A : V ! [−1,1] vs. A : V ! Bn
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 ||.
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]
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
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.
(matches GW for c ¸ .845)
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
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?