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Semidefinite Programming Based Approximation Algorithms

Semidefinite Programming Based Approximation Algorithms. Uri Zwick Tel Aviv University UKCRC’02, Warwick University, May 3, 2002. Outline of talk. Semidefinite programming MAX CUT (Goemans, Williamson ’95 ) MAX 2 -SAT and MAX DI-CUT (FG’95, MM’01, LLZ’02)

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Semidefinite Programming Based Approximation Algorithms

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  1. Semidefinite Programming Based Approximation Algorithms Uri ZwickTel Aviv University UKCRC’02, Warwick University, May 3, 2002.

  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 Algorithms work well in practice, not only in theory!

  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. Pre-rounding rotations(Feige-Goemans ‘95)

  25. Skewed hyperplanes(Feige-Goemans ’95, Matuura-Matsui ’01) Choose a random vector rthat isskewed toward v0. Without loss of generality v0 = (1,0, …,0). Let r = (r1 , r2 ,…, rn), where r2 ,…, rn ~ N(0,1).Choose r1 according to a different distribution.

  26. “Threshold” rounding(Lewin-Livnat-Zwick ’02) Choose a random vector rperpendicular to v0. Set xi=1 iff vi ·r≥ T(v0·vi).

  27. Results for MAX 2-SAT

  28. The MAX 3-SAT problem(Karloff-Zwick ’97 Zwick ’02) A performance ratio of 7/8 is obtained using: • A more complicated SDP relaxation • The simple random hyperplane rounding. • A much more complicated analysis. • Computer assisted proof. (Z’02)

  29. Approximability and Inapproximability results

  30. 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)

  31. (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).

  32. 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.

  33. 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 :

  34. 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.

  35. Constructing a large IS

  36. 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)

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

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