Semidefinite Programming Based Approximation Algorithms Uri Zwick Uri Zwick Tel Aviv University...
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Transcript of Semidefinite Programming Based Approximation Algorithms Uri Zwick Uri Zwick Tel Aviv University...
Semidefinite Programming Semidefinite Programming Based Approximation Based Approximation
AlgorithmsAlgorithms
Uri ZwickUri Zwick
Tel Aviv University
UKCRC’02, Warwick UKCRC’02, Warwick University, University,
May 3, 2002.May 3, 2002.
Outline of talkOutline 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 MatricesPositive Semidefinite Matrices
A symmetric nn matrix A is PSDPSD iff:
• xTAx 0 , for every xRn.
• A=BTB , for some mn matrix B.
• All the eigenvalues of A are non-negative.
Notation: A 0 iff A is PSD
Linear Linear ProgrammiProgrammi
ngngmax c x
s.t. ai x bi
x 0
Semidefinite Semidefinite ProgrammingProgramming
max CX
s.t. Ai X bi
X 0
Can be solved exactly
in polynomial time
Can be solved almost exactly
in polynomial time
LP/SDP algorithmsLP/SDP algorithms
• Simplex method (LP only)
• Ellipsoid method
• Interior point methods
Algorithms work well in practice, not only in theory!
Semidefinite Semidefinite ProgrammingProgramming(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 Lovász’s -function-function(one of many formulations)
max JX
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 TheoremThe Sandwich Theorem(Grötschel-Lovász-Schrijver ’81)
)G()G()G(
Size of max clique
Chromaticnumber
The The MAX CUTMAX CUT problem: problem:
motivationmotivation 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 BISECTIONMAX BISECTION..
The The MAX CUTMAX CUT problem: problem: statusstatus
• 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 A quadratic integer programming formulation of programming formulation of
MAX CUTMAX CUT
}1,1{ s.t.2
1Max
i
jiij
x
xxw
An SDP Relaxation of An SDP Relaxation of MAX MAX CUTCUT
(Goemans-Williamson ’95)
1||||, s.t.
2
1Max
in
i
jiij
vRv
vvw
An SDP Relaxation of An SDP Relaxation of MAX CUT – MAX CUT –
Geometric intuitionGeometric intuition
Embed the vertices of the graph on the unit sphere such that vertices that are joined by edges are far apart.
Random hyperplane Random hyperplane roundingrounding
(Goemans-Williamson ’95)(Goemans-Williamson ’95)
To choose a random hyperplane,
choose a random normal vector
r
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.
Analysis of the Analysis of the MAX CUTMAX CUT Algorithm Algorithm (Goemans-Williamson (Goemans-Williamson
’95)’95)
1
1
1
1
exp
sdp
ratio min
1
2
2
1
cos
0.8785
( )
cos (.
)6. .
ii
jj
ijj
x
i
w
v v
x
v
w
v
x
A Semidefinite A Semidefinite Programming Relaxation Programming Relaxation
of of MAX 2-SATMAX 2-SAT(Feige-Lovász ’92, Feige-Goemans ’95)
1
1
201s.t.4
3Max 00
||||,
,
,,,
in
i
iin
kjkiji
jijiij
vRv
nivv
nkjivvvvvv
vvvvvvw
Triangle constraints
Pre-rounding Pre-rounding rotationsrotations
(Feige-Goemans ‘95)iv
0v
iv
iv
i0 ivi0
)()(
],0[],0[:
)( 00
ff
f
f ii
Skewed hyperplanesSkewed 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” roundingThreshold” rounding(Lewin-Livnat-Zwick ’02)
Choose a random vector r perpendicular to v0.
Set xi=1 iff vi · r ≥ T( v0· vi ).
Results for Results for MAX 2-SATMAX 2-SATAuthorsTechniqueBound
Goemans-Williamson ‘95Random
hyperplane0.878
Feige-Goemans ‘95Pre-rounding
rotations0.931
Matuura-Matsui ‘01Skewed
hyperplanes0.935
Lewin-Livnat-Zwick ‘02Threshold rounding0.941
Integrality ratio *0.945
Inapproximability0.954
The The MAX 3-SATMAX 3-SAT problem problem(Karloff-Zwick ’97 Zwick ’02)(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 Approximability and Inapproximability resultsInapproximability results
ProblemApprox.
RatioInapprox.
RatioAuthors
MAX CUT0.87816/17 0.941
Goemans Williamson ’95
MAX DI-CUT0.87412/13 0.923
GW’95, FW’95 MM’01, LLZ’01
MAX 2-SAT0.94121/22 0.954
GW’95, FW’95 MM’01, LLZ’01
MAX 3-SAT7/87/8Karloff
Zwick ’97
What else can we What else can we do with do with SDPSDPs?s?
• MAX BISECTION MAX BISECTION (Frieze-Jerrum ’95)
• MAX MAX kk-CUT-CUT (Frieze-Jerrum ’95)
• (Approximate) Graph colouring (Karger-Motwani-Sudan’95)
(Approximate) Graph (Approximate) Graph colouringcolouring
• 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 Vector kk-Coloring-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 Finding large independent setssets
((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.
lnlnln 31
32c
}|{ crvViI i
Colouring Colouring kk-colourable -colourable graphsgraphs
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)