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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 ( Lov á sz ’79)

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semidefinite programming

Lecture 8:

Semidefinite Programming

Magnus M. Halldorsson

Based on slides by Uri Zwick

outline of talk
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
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

linear programming

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

lp sdp algorithms
LP/SDP algorithms
  • Simplex method (LP only)
  • Ellipsoid method
  • Interior point methods
semidefinite programming equivalent formulation
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.

lov sz s function one of many formulations
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 max cut problem
The MAX CUT problem

Edges may be weighted

the max cut problem motivation
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
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)
an sdp relaxation of max cut geometric intuition
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.

to choose a random hyperplane choose a random normal vector

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.

is the analysis tight
Is the analysis tight?

Yes!

(Karloff ’96) (Feige-Schechtman ’00)

the max directed cut problem
The MAX Directed-CUT problem

Edges may be weighted

a semidefinite programming relaxation of max 2 sat feige lov sz 92 feige goemans 95

Triangle constraints

A Semidefinite Programming Relaxation of MAX 2-SAT(Feige-Lovász ’92, Feige-Goemans ’95)
what else can we do with sdp s
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
(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

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.

slide28

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 :

finding large independent sets karger motwani sudan 95

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.

colouring k colourable graphs
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

Open problems

Improved results for the problems considered.

Further applications of SDP.