Approximating the Cut-Norm

1. Approximating the Cut-Norm Hubert Chan

2. “Approximating the Cut-Norm via Grothendieck’s Inequality” Noga Alon, Assaf Naor appearing in STOC ‘04

3. _ _ _ ++++ + _ + + + + _ _ _ + _ Problem Definition

4. Main Result • The problem is MAX SNP hard. • Randomized polynomial algorithm that gives expected 0.56-approximation. For maximization problem, approximation ratio always less than 1. The authors showed a deterministic algorithm that gives 0.03-approximation. De-randomization: paper by Mahajan and Ramesh

5. Road Map • Motivation • Hardness Result • General Approach • Outline of Algorithm • Conclusion

6. Motivation • Inspired by the MAX-CUT problem Frieze and Kannan proposed decomposition scheme for solving problems on dense graphs • Estimating the norm of a matrix is a key step in the decomposition scheme

7. Comparing with Previous result • Previously, computes norm with additive error • This is good only for a matrix whose norm is large. • The new algorithm approximates norm for all real matrices within constant factor 0.56 in expectation.

8. Road Map • Motivation • Hardness Result • General Approach • Outline of Algorithm • Conclusion

9. MAX-SNP A maximization problem is MAX-SNP hard if For example, there is a well-known polynomial algorithm for MAX-CUT that returns a cut with size at least 0.5 of the maximum cut. However, there is no polynomial algorithm that gives 16/17-approximation.

10. MAX-CUT Graph G=(V,E) W V\W

11. v e u u v e,1 1/4 -1/4 e,2 -1/4 1/4 The problem is MAX SNP hard • Reduction from MAX-CUT • Given graph G = (V,E), construct 2|E| x |V| matrix A: for each edge e = (u,v),

12. u v 1/4 -1/4 _ + u v + 1/4 -1/4 _ -1/4 1/4 MAX-CUT · ||A||§ For e = (u,v) not in max cut, there is no contribution no matter what xe,1 and xe,2 are. For e = (u,v) in max cut, we can set xe,1 and xe,2 to give contribution 1.

13. u v 1/4 -1/4 _ + u v + 1/4 -1/4 _ -1/4 1/4 MAX-CUT ¸ ||A||§ For e = (u,v) not in cut (W,V\W), there is no contribution no matter what xe,1 and xe,2 are. For e = (u,v) in cut (W, V\W), the contribution from rows e,1 and e,2 is at most 1.

14. Road Map • Motivation • Hardness Result • General Approach • Outline of Algorithm • Conclusion

15. Relaxation Schemes • Recall the problem: Linear Programming Relaxation? • Objective function not linear • Could introduce extra variables, but rounding might be tricky • How about Semidefinite Program Relaxation?

16. Semidefinite Program Relaxation ||A||SDP = max ij aij ui vj ui² ui = 1 vj² vj = 1 where ui and vj are vectors in m+n dimensional Euclidean space

17. Remarks about SDP ² Are (m+n)-dimensions sufficient? Yes, since any m+n vectors in a higher dimensional Euclidean space lie on an (m+n)-dimensional subspace. ² Fact: There exists an algorithm that given  > 0, returns solution vectors ui’s and vj’s that attains value at least ||A||SDP -  in time polynomial in the length of input and the logarithm of 1/.

18. Are we done? We need to convert the vectors back to integers in {-1,1}! General strategy: 1. Obtain optimal vectors ui and vj for the SDP. 2. Use some randomized procedure to reconstruct integer solutions xi, yj2 {-1,1} from the vectors. • Give good expected bound:Find some constant  > 0 such that • E[ij aij xi yj] ¸ ||A||SDP¸ ||A||§

19. Road Map • Motivation • Hardness Result • General Approach • Outline of Rounding Algorithm • Conclusion

20. z Random Hyperplane + _ Recall we need to show: E[ij aij xi yj] ¸ij aij ui²vj

21. u q v z Analyzing E[xy] Unit vectors u and v such that cos  = u²v A random unit vector z determines a hyperplane. Pr[u and v are separated] =  /  Set x = sign(u²z), y = sign(v²z). E[xy] = (1 -  / ) -  /  = 1 - 2  /  = 2/  (  / 2 - ) = 2/  arcsin(u²v)

22. How do sine and arcsine relate? Is this good news?

23. Performance Guarantee? • We have term by term constant factor approximation. • Bad news: cancellation because terms have different signs • Hence, we need global approximation.

24. Generate standard, independent Gaussian random variables r1, r2, …, rm+n _ An Equivalent Way to Round Vectors + R R = (r1, r2, …, rm+n) Set xi= sign(ui²R), yj = sign(vj²R)

25. What we would like to see…. This is impossible because arcsin is not a linear function.

26. What we can prove…… where fi is a function depending on ui and gj is a function depending on vj. Important property of fi and gj: E[fi2] = E[gj2] =  /2 – 1 < 1.

27. Inner Product and E[f g]

28. Recall the SDP ||A||SDP = max ij aij ui vj ui² ui = 1 vj² vj = 1 where ui and vj are vectors in m+n dimensional Euclidean space Are (m+n)-dimensions sufficient? Yes, since any m+n vectors in a higher dimensional Euclidean space lie on an (m+n)-dimensional subspace.

29. Wait a minute… We need unit vectors!

30. Constant factor approximation

31. What are functions f and g?

32. Properties of Gaussian Measure (a) Mean 0, Variance 1 (b) Multi-dimensional Gaussian spherical symmetric

33. 3. Relate E[xi yj] to ui²vj. Recap 1. Solve for optimal vectors ui and vj for the SDP. 2. Generate multi-dimensional Gaussian random vector R. Set xi = sign(ui²R), yj = sign(vj² R). 4. Use (1) ui and vj are optimal vectors and (2) E[fi gj] can be considered as an inner product. E[ij aij xi yj] ¸ 0.273 ||A||§

34. What we would like to see…. This is impossible because arcsin is not a linear function.

35. What if…

36. If this is possible…. Recall z is the random unit vector.

37. This is indeed possible!

38. Another Semidefinite Program

39. Better Constant Approximation

40. Road Map • Motivation • Hardness Result • General Approach • Outline of Algorithm • Conclusion

41. Main Ideas • Semidefinite Program Relaxation - a powerful tool for optimization problems • Randomized Rounding Scheme - random hyperplane - multi-dimensional Gaussian • Apply similar techniques directly to approximate MAX-CUT