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Subhash Khot Georgia Tech

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]

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Subhash Khot Georgia Tech

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  1. SDP Gaps and UGC-Hardness for Max-Cut-Gain Subhash KhotGeorgia Tech Ryan O’DonnellCarnegie Mellon &

  2. 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

  3. [Trivial algorithm] [Karp’72]:5/6vs.5/6 − 1/poly(n)NP-hard [Sahni-Gonzalez’76] [Goemans-Williamson’95]: .878 factor [Håstad+TSSW’97]: 17/21vs.16/21NP-hard [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

  4. 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 + 

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

  6. 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))

  7. 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

  8. 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))

  9. 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))

  10. 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]

  11. 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:

  12. 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]

  13. 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]

  14. 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]

  15. 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.

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

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