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Semidefinite Programming Gaps in Weighted Graphs: UGC-Hardness & Max-Cut Gain

Explore the intersection of semidefinite programming integrality gaps, UGC-hardness, and the Max-Cut Gain problem in weighted graphs. Discover the complexities and implications of different assignments and compare various approaches to cutting graphs efficiently in polynomial time. Uncover the nuances of Gaussian space translations and code tests in this comprehensive study.

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Semidefinite Programming Gaps in Weighted Graphs: UGC-Hardness & Max-Cut Gain

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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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