Presentation is loading. Please wait.

# Subhash Khot Georgia Tech Ryan O’Donnell Carnegie Mellon SDP Gaps and UGC-Hardness for Max-Cut-Gain &

## Presentation on theme: "Subhash Khot Georgia Tech Ryan O’Donnell Carnegie Mellon SDP Gaps and UGC-Hardness for Max-Cut-Gain &"— Presentation transcript:

Subhash Khot Georgia Tech Ryan O’Donnell Carnegie 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 A c. A.059.183.097

When OPT is c, can you in poly-time cut s ? c s 1 1/2 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 ! B n 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 vs. 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 ! B n 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 vs. 1. Pick G, rand. n-dim. Gaussian 2. Define A(x) = sgn(G ¢ A(x))2. Define A(x) = F (G ¢ A(x)) F 1 −1−1

Semidefinite programming gaps Weighted graph:H = (V, w : V £ V ! R ¸ 0 ) Assignments:A : V ! [−1,1] vs. A : V ! B n 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 vs. 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 ! B n 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 vs. 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 ! B n 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 vs. 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 = R n, w = picking (1−2c)-correlated Gaussians. (matches GW for c ¸.845) 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 ! B n 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 vs. Take A(x) = x / || x ||. This paper: “There exists H s.t. s · blah(c).” Proof: Take V = R n, w = picking (1−2c)-correlated Gaussians. (essentially matches FL/CW for c = 1/2 +  ) Proof: Take V = R n, 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 ! B n 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 vs. 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 = R n, 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 vs. 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 = R n, 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 ! B n Compare: Weighted graph:H = ({−1,1} n, w : V £ V ! R ¸ 0 ) Assignments:A : {−1,1} n ! [−1,1] vs. A i (x) = x i far from all Dictators i ii

c : = max E [ (½) − (½) A(x) ¢ A(y) ] s : = max E [ (½) − (½) A(x) ¢ A(y) ] A(x,y) Ã w vs. 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 = R n, 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. A i (x) = x i far from all Dictators i ii 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 vs. (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. A i (x) = x i far from all Dictators i ii 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?

Download ppt "Subhash Khot Georgia Tech Ryan O’Donnell Carnegie Mellon SDP Gaps and UGC-Hardness for Max-Cut-Gain &"

Similar presentations

Ads by Google