Semidefinite Programming Based Approximation Algorithms Uri Zwick 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) MAX 3-SAT (Karloff, Zwick ’97) -function (Lovász ’79) MAX k-CUT (Frieze, Jerrum ’95) Colouring k-colourable graphs (Karger, Motwani, Sudan ’95)

Positive Semidefinite Matrices PSD A symmetric n  n matrix A is PSD iff: x T Ax  0, for every x  R n. A=B T B, for some m  n matrix B. All the eigenvalues of A are non-negative. Notation: A  0 iff A is PSD

Linear Programming max c  x s.t. a i  x  b i x  0 Semidefinite Programming max C  X s.t. A i  X  b i X  0 Can be solved exactly in polynomial time Can be solved almost exactly in polynomial time

LP/SDP algorithms Simplex method (LP only) Ellipsoid method Interior point methods Algorithms work well in practice, not only in theory!

Semidefinite Programming Semidefinite Programming (Equivalent formulation) max  c ij (v i  v j ) s.t.  a ij (k) (v i  v j )  b (k) v i  R n X ≥ 0 iff X=B T B. If B = [v 1 v 2 … v n ] then x ij = v i · v j.

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

The Sandwich Theorem The Sandwich Theorem (Grötschel-Lovász-Schrijver ’81) Size of max clique Chromatic number

The MAX CUT problem Edges may be weighted

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. MAX BISECTION. If exactly n/2 of the activities have to be held in the morning, we get MAX BISECTION.

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)

A quadratic integer programming formulation of MAX CUT

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

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.

Random hyperplane rounding (Goemans-Williamson ’95)

To choose a random hyperplane, choose a random normal vector r If r = (r 1, r 2, …, r n ), and r 1, r 2, …, r n  N(0,1), then the direction of r is uniformly distributed over the n-dimensional unit sphere.

The probability that two vectors are separated by a random hyperplane vivi vjvj

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

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

The MAX Directed-CUT problem Edges may be weighted

The MAX 2-SAT problem

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

The probability that a clause x i  x j is satisfied is :

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

Skewed hyperplanes Skewed hyperplanes (Feige-Goemans ’95, Matuura-Matsui ’01) Choose a random vector r that is skewed toward v 0. Without loss of generality v 0 = (1,0, …,0). Let r = (r 1, r 2, …, r n ), where r 2, …, r n ~ N(0,1). Choose r 1 according to a different distribution.

“Threshold” rounding “Threshold” rounding (Lewin-Livnat-Zwick ’02) Choose a random vector r  perpendicular to v 0. Set x i =1 iff v i · r  ≥ T( v 0 · v i ).

Results for MAX 2-SAT AuthorsTechniqueBound Goemans-Williamson ‘95 Random hyperplane Feige-Goemans ‘95 Pre-rounding rotations Matuura-Matsui ‘01 Skewed hyperplanes Lewin-Livnat-Zwick ‘02 Threshold rounding Integrality ratio * Inapproximability 0.954

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)

Approximability and Inapproximability results Problem Approx. Ratio Inapprox. Ratio Authors MAX CUT / Goemans Williamson ’95 MAX DI-CUT / GW’95, FW’95 MM’01, LLZ’01 MAX 2-SAT / GW’95, FW’95 MM’01, LLZ’01 MAX 3-SAT7/8 Karloff Zwick ’97

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

(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 n 1/2 colours. (Wigderson’81) Using SDP, can colour (in poly. time) using n 1/4 colours (KMS’95), or even n 3/14 colours (BK’97).

Vector k -Coloring ( Vector k -Coloring (Karger-Motwani-Sudan ’95) A vector k-coloring of a graph G = (V,E) is a sequence of unit vectors v 1, v 2, …, v n such that if (i,j)  E then v i · v j = -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.

Lemma: If G = (V,E) is k-colorable, then it is also vector k-colorable. Proof: There are k vectors v 1,v 2, …, v k such that v i · v j = -1/(k-1), for i ≠ j. k = 3 :

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

Constructing a large IS

Colouring k-colourable graphs Colouring k-colourable graphs using min{ Δ 1-2/k, n 1-3/(k+1) } colours. (Karger-Motwani-Sudan ’95) Colouring 3-colourable graphs using n 3/14 colours. (Blum-Karger ’97) Colouring 4-colourable graphs using n 7/19 colours. (Halperin-Nathaniel-Zwick ’01)

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