Semidefinite Programming

1. Lecture 8: Semidefinite Programming Magnus M. Halldorsson Based on slides by Uri Zwick

2. Outline of talk Semidefinite programming MAX CUT (Goemans, Williamson ’95) MAX 2-SAT and MAX DI-CUT (FG’95, MM’01, LLZ’02) MAX 3-SAT (Karloff, Zwick ’97) -function (Lovász ’79) MAX k-CUT (Frieze, Jerrum ’95) Colouring k-colourable graphs (Karger, Motwani, Sudan ’95)

3. Positive Semidefinite Matrices A symmetric nnmatrix A is PSDiff: • xTAx  0, for every xRn. • A=BTB , for some mnmatrix B. • All the eigenvalues of A are non-negative. Notation: A0iff A is PSD

4. Semidefinite Programming Linear Programming max CX s.t. AiX bi X  0 max cx s.t. ai x  bi x  0 Can be solved exactlyin polynomial time Can be solvedalmost exactlyin polynomial time

5. LP/SDP algorithms • Simplex method (LP only) • Ellipsoid method • Interior point methods

6. Semidefinite Programming(Equivalent formulation) max  cij(vi vj) s.t.  aij(k)(vi vj) b(k) viRn X ≥ 0 iff X=BTB. If B = [v1 v2 … vn] then xij = vi ·vj.

7. Lovász’s -function(one of many formulations) max JX s.t. xij =0 , (i,j)E IX= 1 X  0 Orthogonal representation of a graph: vi vj =0 , whenever (i,j)E

8. The Sandwich Theorem(Grötschel-Lovász-Schrijver ’81) Size of max clique Chromaticnumber

9. The MAX CUT problem Edges may be weighted

10. The MAX CUT problem: motivation Given: n activities, m persons. Each activity can be scheduled either in the morning or in the afternoon. Each person interested in two activities. Task: schedule the activities to maximize the number of persons that can enjoy both activities. If exactly n/2 of the activities have to be held in the morning, we get MAX BISECTION.

11. The MAX CUT problem: status • Problem is NP-hard • Problem is APX-hard (no PTAS unless P=NP) • Best approximation ratio known, without SDP, is only ½. (Choose a random cut…) • With SDP, an approximation ratio of 0.878 can be obtained! (Goemans-Williamson ’95) • Getting an approximation ratio of 0.942is NP-hard! (PCP theorem, …, Håstad’97)

12. A quadratic integer programming formulation of MAX CUT

13. An SDP Relaxation of MAX CUT(Goemans-Williamson ’95)

14. An SDP Relaxation of MAX CUT – Geometric intuition Embed the vertices of the graph on the unit spheresuch that vertices that are joined by edges are far apart.

15. Random hyperplane rounding(Goemans-Williamson ’95)

16. r To choose a random hyperplane,choose a random normal vector Ifr = (r1 , r2 , …, rn),andr1, r2 , … , rn  N(0,1),thenthe direction of ris uniformly distributed over the n-dimensional unit sphere.

17. The probability that two vectors are separated by a random hyperplane vi vj

18. Analysis of the MAX CUT Algorithm (Goemans-Williamson ’95)

19. Is the analysis tight? Yes! (Karloff ’96) (Feige-Schechtman ’00)

20. The MAX Directed-CUT problem Edges may be weighted

21. The MAX 2-SAT problem

22. Triangle constraints A Semidefinite Programming Relaxation of MAX 2-SAT(Feige-Lovász ’92, Feige-Goemans ’95)

23. The probability that a clause xi xj is satisfied is :

24. Approximability and Inapproximability results

25. What else can we do with SDPs? • MAX BISECTION (Frieze-Jerrum ’95) • MAX k-CUT(Frieze-Jerrum ’95) • (Approximate) Graph colouring(Karger-Motwani-Sudan’95)

26. (Approximate) Graph colouring • Given a 3-colourable graph, colour it, in polynomial time, using as few colours as possible. • Colouring using 4 colours is still NP-hard. (Khanna-Linial-Safra’93 Khanna-Guruswami’01) • A simple combinatorial algorithm can colour, in polynomial time, using about n1/2 colours.(Wigderson’81) • Using SDP, can colour (in poly. time) using n1/4 colours (KMS’95), or even n3/14 colours (BK’97).

27. Vector k-Coloring(Karger-Motwani-Sudan ’95) A vector k-coloring of a graph G = (V,E) is a sequence of unit vectors v1 , v2 , … , vn such that if (i,j)E then vi ·vj = -1/(k-1). The minimum k for which G is vector k-colorable is A vector k-coloring, if one exists, can be found using SDP.

28. Lemma: If G = (V,E)is k-colorable, then it is also vectork-colorable. Proof: There are k vectors v1 ,v2 , … , vk such that vi ·vj = -1/(k-1), for i ≠ j. k = 3 :

29. Finding large independent sets(Karger-Motwani-Sudan ’95) Let r be a random normally distributed vector inRn. Let . I’ is obtained from I by removing a vertex from each edge ofI.

30. Constructing a large IS

31. Colouring k-colourable graphs Colouring k-colourable graphs using min{ Δ1-2/k,n1-3/(k+1) } colours.(Karger-Motwani-Sudan ’95) Colouring 3-colourable graphs using n3/14 colours. (Blum-Karger ’97) Colouring 4-colourable graphs using n7/19 colours. (Halperin-Nathaniel-Zwick ’01)

32. Open problems Improved results for the problems considered. Further applications of SDP.