Semidefinite Programming

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Semidefinite Programming
Lecture 8
Magnus M. Halldorsson

• Based on slides by Uri Zwick

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Outline of talk
Semidefinite programming MAX CUT (Goemans,
Williamson 95) MAX 2-SAT and MAX DI-CUT (FG95,
MM01, LLZ02) MAX 3-SAT (Karloff, Zwick
97) ?-function (Lovász 79) MAX k-CUT
(Frieze, Jerrum 95) Colouring k-colourable
graphs (Karger, Motwani, Sudan 95)
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Positive Semidefinite Matrices

• A symmetric n?n matrix A is PSD iff
• xTAx ? 0 , for every x?Rn.
• ABTB , for some m?n matrix B.
• All the eigenvalues of A are non-negative.
• Notation A ? 0 iff A is PSD

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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 exactlyin polynomial time
Can be solved almost exactlyin polynomial time
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LP/SDP algorithms

• Simplex method (LP only)
• Ellipsoid method
• Interior point methods

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Semidefinite Programming(Equivalent formulation)
max ? cij (vi? vj) s.t. ? aij(k) (vi? vj) ?
b(k) vi ? Rn
X 0 iff XBTB. If B v1 v2 vn then
xij vi vj .
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Lovászs ?-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
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The Sandwich Theorem(Grötschel-Lovász-Schrijver
81)
Size of max clique
Chromaticnumber
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The MAX CUT problem
Edges may be weighted
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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.
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The MAX CUT problem status

• Problem is NP-hard
• Problem is APX-hard (no PTAS unless PNP)
• 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, , Håstad97)

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A quadratic integer programming formulation of
MAX CUT
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An SDP Relaxation of MAX CUT(Goemans-Williamson
95)
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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.
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Random hyperplane rounding(Goemans-Williamson
95)
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To choose a random hyperplane,choose a random
normal vector
If r (r1 , r2 , , rn), andr1, r2 , , rn ?
N(0,1), then the direction of r is uniformly
distributed over the n-dimensional unit sphere.
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The probability that two vectors are separated
by a random hyperplane
vi
vj
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Analysis of the MAX CUT Algorithm
(Goemans-Williamson 95)
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Is the analysis tight?
Yes! (Karloff 96) (Feige-Schechtman 00)
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The MAX Directed-CUT problem
Edges may be weighted
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The MAX 2-SAT problem
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A Semidefinite Programming Relaxation of MAX
2-SAT(Feige-Lovász 92, Feige-Goemans 95)
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The probability that a clause xi ? xj is
satisfied is
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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 GW95, FW95 MM01, LLZ01
MAX 2-SAT 0.941 21/22 0.954 GW95, FW95 MM01, LLZ01
MAX 3-SAT 7/8 7/8 Karloff Zwick 97
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What else can we do with SDPs?

• MAX BISECTION (Frieze-Jerrum 95)
• MAX k-CUT (Frieze-Jerrum 95)
• (Approximate) Graph colouring
  (Karger-Motwani-Sudan95)

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(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-Safra93 Khanna-Guruswami01)
• A simple combinatorial algorithm can colour, in
  polynomial time, using about n1/2 colours.
  (Wigderson81)
• Using SDP, can colour (in poly. time) using n1/4
  colours (KMS95), or even n3/14 colours (BK97).

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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.
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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
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Finding large independent sets(Karger-Motwani-Sud
an 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.

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Constructing a large IS
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Colouring k-colourable graphs
Colouring k-colourable graphs using min ?1-2/k
, n1-3/(k1) 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)
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Open problems

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