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

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 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

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

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

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 .

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

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. 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 0.878 can be obtained! (Goemans-Williamson ’95) Getting an approximation ratio of 0.942 is NP-hard! (PCP theorem, …, Håstad’97)

A quadratic integer programming formulation 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 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

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

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

(Karloff ’96) (Feige-Schechtman ’00) 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 (Feige-Lovász ’92, Feige-Goemans ’95) Triangle constraints

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

Approximability and Inapproximability results Problem Approx. Ratio Inapprox. Ratio Authors MAX CUT 0.878 16/17 0.941 Goemans Williamson ’95 MAX DI-CUT 0.874 12/13 0.923 GW’95, FW’95 MM’01, LLZ’01 MAX 2-SAT 0.941 21/22 0.954 MAX 3-SAT 7/8 Karloff Zwick ’97

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)

(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).

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.

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 :

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.

Constructing a large IS

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)

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