1. Semidefinite Programming
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n matrix A is PSD iff:
xTAx  0 , for every xRn.
A=BTB , for some mn matrix B.
All the eigenvalues of A are non-negative.
Notation: A  0 iff A is PSD

4. Semidefinite Programming
Linear Programming
Semidefinite Programming
max c x
s.t. ai x  bi
x  0
max CX
s.t. Ai X  bi
X  0
Can be solved exactly in polynomial time
Can be solved almost exactly in 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
Chromatic number

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 can be obtained! (Goemans-Williamson ’95)
Getting an approximation ratio of is 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 sphere such that vertices that are joined by edges are far apart.

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

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

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

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

19. (Karloff ’96) (Feige-Schechtman ’00)
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. A Semidefinite Programming Relaxation of MAX 2-SAT (Feige-Lovász ’92, Feige-Goemans ’95)
Triangle constraints

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

24. Approximability and Inapproximability results
Problem
Approx. Ratio
Inapprox. Ratio
Authors
MAX CUT
0.878
16/
Goemans Williamson ’95
MAX DI-CUT
0.874
12/
GW’95, FW’95 MM’01, LLZ’01
MAX 2-SAT
0.941
21/
MAX 3-SAT
7/8
Karloff Zwick ’97

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 vector k-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 in Rn. Let
I’ is obtained from I by removing a vertex from each edge of I.

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.