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

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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

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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

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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.

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Theme of the paper: Semidefinite programming integrality gaps arise naturally in Gaussian space. Can be translated into Long Code tests; ) UGC-hardness.

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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))

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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

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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))

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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:

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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]

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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.

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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]

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(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

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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”)

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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.

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

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