Download Semidefinite Programming Based Approximation Algorithms

Semidefinite Programming
Based Approximation
Algorithms
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
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
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
Algorithms work well in practice,
not only in theory!
Semidefinite Programming
(Equivalent formulation)
max  cij (vi vj)
(k)
(k)
s.t.  aij (vi vj)  b
v i  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)
 (G )   ( G )   (G )
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
Max  wij
s.t.
1  xi x j
2
xi {1,1}
An SDP Relaxation of MAX CUT
(Goemans-Williamson ’95)
Max
s.t.
w
ij
1  vi  v j
2
n
vi  R , || vi || 1
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 = (r1 , r2 , …, rn), and
r1, r2 , … , rn  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
vi

vj


Analysis of the MAX CUT Algorithm
(Goemans-Williamson ’95)
exp 
sdp 
w
1
cos (vi  v j )

ij
w
1  vi  v j
ij
1
2
2 cos ( x )
ratio  min
 0.87856...
1 x 1 
1 x
Is the analysis tight?
Yes!
(Karloff ’96)
(Feige-Schechtman ’00)
The MAX Directed-CUT problem
Edges may be weighted
The MAX 2-SAT problem
x1  x 2
x 2  x5
x3  x4
x2  x6
x3  x5
x3  x4
x4  x5
x3  x7
x4  x7
A Semidefinite Programming
Relaxation of MAX 2-SAT
(Feige-Lovász ’92, Feige-Goemans ’95)
Max
s.t.
w
3  v0  v i  v0  v j  v i  v j
ij
4
v i  v j  v i  v k  v j  v k   1 , 0  i , j , k  2n
v n  i  v i
, 1 i  n
v i  R , || v i || 1
n
Triangle constraints
The probability that a
clause xi  xj is satisfied is :
v0
0i  0 j  ij
2
vi
 ij
vj
Pre-rounding rotations
(Feige-Goemans ‘95)
vi
v0
 0i
 v i
 vi
 0i
vi
 0i  f ( 0i )
f : [0,  ]  [0,  ]
f (   )    f ( )
Skewed hyperplanes
(Feige-Goemans ’95, Matuura-Matsui ’01)
Choose a random vector r
that is skewed toward v0.
Without loss of generality v0 = (1,0, …,0).
Let r = (r1 , r2 , …, rn), where r2 , …, rn ~ N(0,1).
Choose r1 according to a different distribution.
“Threshold” rounding
(Lewin-Livnat-Zwick ’02)
Choose a random vector r
perpendicular to v0.
Set xi=1 iff vi · r ≥ T( v0· vi ).
Results for MAX 2-SAT
Authors
Technique
Goemans-Williamson ‘95
Random
hyperplane
Pre-rounding
rotations
Skewed
hyperplanes
Threshold
rounding
Feige-Goemans ‘95
Matuura-Matsui ‘01
Lewin-Livnat-Zwick ‘02
Bound
0.878
0.931
0.935
0.941
Integrality ratio *
0.945
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
MAX CUT
MAX DI-CUT
MAX 2-SAT
MAX 3-SAT
Approx.
Ratio
Inapprox.
Ratio
Authors
0.878
16/17
0.941
Goemans
Williamson ’95
0.874
12/13
0.923
GW’95, FW’95
MM’01, LLZ’01
0.941
21/22
0.954
GW’95, FW’95
MM’01, LLZ’01
7/8
Karloff
Zwick ’97
7/8
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  (G )
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 c  23 ln   13 ln ln  .
I  {i V | vi  r  c}
I’ is obtained from I by removing a
vertex from each edge of I.
Constructing a large IS
vi
r
vj
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.