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Semi-Definite Algorithm for Max-CUT
Ran BerenfeldMay 10,2005
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Agenda :
•The Max-Cut problem.•Goemans-Williamson algorithm.•Semi-Definite programming.•Other applications.
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The Max-Cut Problem :
Let be a complete, undirected graph,
With edge weights .
Find a cut that maximizes
EVG ,
QEW :
S
SwSv
wve
ew
,,
)(
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•Observations :
•General definition •set weight=1 if edges are un-weighted.•set weight=0 for non complete graph.
•NP-Hard [Karp 72’]• approximation is easy.•This presentation – [Goemans-Williamson 94’]shows -approximation where
•[Karloff ’99, Feige-Schechtman ’99] – Goemans Williamson have an integralitty gap of
21
...878.0cos1
2min0
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GW strategy for Max-Cut
Graph
QPVP
SDP
1.Write problem as a Quadratic Problem. (with integer solutions)2.Relax to vector programming. 3.Vector programming is equal to semi-definite programming (SDP).4.Solve SDP.
Approx
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Graph
QP
Assign a variable to each vertex.Let for vertices in Let for vertices in
ix1ix S
1ix S
}1,1{.
)1(2
1max ,
i
jijiji
xts
xxw
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QPVP
Replace each with .jixx ji yy ,Old objective value is achieved setting where where
)0,...,0,0,1(iy 1ix)0,...,0,0,1(iy 1ix
1.
),1(2
1max ,
i
jijiji
yts
yyw
Approx
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QPVP
Approx
Motivation : heavy weighted verticeswill be “far” away from each other.
1000
iv jv
iy
0,1 ji yy
jy
jy
1,1 ji yy
2,1 ji yy jy
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VPSDP
we’ll show later that VP is equal to SDP.
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SDP
we’ll also show later how SDP is polynomial time solvable to any accuracy degree.But first lets analyze the approximation ratio.
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Suppose are the vectors solution to our VP.To obtain a cut from the solution : Randomly pick a vector on the unit sphere, and let
nvv ,...,1
SDP
r}0,|{ rvvS ii
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Let and be vectors in the VP solution.iv jv
By the choice of it follows thatPr[the edge is in the cut]=Pr[ ]),(),( rvsignrvsign ji
S ji,
And so the expected weight of the cut produced by the algorithm is :
ji
jiji rvsignrvsignwWE )],(),(Pr[][ ,
Approximation Analysis :
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),arccos(
)],(),(Pr[
jiji
vvrvsignrvsign
If the angle between and is , there is an area of size where can satisfy
iv jv 2 r
),(),( rvsignrvsign ji
iv
jv
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ji
jiji vvwWE ),arccos(1
][ ,
Current conclusion :
The optimal solution to VP is no less then the optimal cut. So it follows :
ji
jiji vvwOPT ),1(2
1,
Now we set
And obtain : !
cos1min
2
0
OPTWE ][
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QPSDP
Integralitty gap :
0 1
VP feasible solution and fractional OPT
OPT-F
0 1
QP solutions and the optimal solution
OPT
0 1
Find integral solution of cost
OPT-F
OPTOPTF
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SDP
A real, symmetric matrix is positive semi-definite if (TFAE) :
1. for all x.2.all eigenvalues of are non
negative.3.there exist a matrix so that
.
A
0AxxT
AB BBA T
0ANotations: means is positive semiDefinite. is the convex of all symmetricMatrices.
A
nM
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SDP
Define (Frobenius product) :
.
n
i
n
jjiji
T baBAtrBA1 1
,,)(
n
ii
MZZ
kidZDts
ZCMax
,0
)..1(.
Where and all ‘s are symmetric.C D
Then SDP in general form is :
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VPSDP
1.Replace with .2.Demand that the matrix beSymmetric and positive semi-definite.
ji yy , jiz ,
}{ , jizZ
It follows that both problems (VP and SDP) are equal.
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SDP
It’s easy to show that SDP can be solved in polynomial time using the Ellipsoid method.Other methods exists that are much more practical…
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SDP
The Ellipsoid methodA convex set in is described using a set of restrictions
nRP
We need to find a point in the set.),,( mnxm RbRAbAxP
We need to be able, for each point To provide a separating hyperplane (in polynomial time)
Py
HPHy ,
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SDP
The Ellipsoid methodThe method starts with a large ellipsoid containing .PAt each step, if the current point is not in ,we use the separating hyperplane to find a (significantlly) smaller ellipsoid.
0x
P1x
P
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SDP
The SDP Problem :
n
ii
MZZ
kidZDts
ZCMax
,0
)..1(.
We treat the matrix as a vector in .Z2nR
The set of symmetric ,positiveSemi-definite matrices is convex.
It follows the set of feasible solution is convex.
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SDP
The SDP Problem :
n
ii
MZZ
kidZDts
ZCMax
,0
)..1(.
Finding a separating hyperplane :
If is not symmetric, is a S.HZ ijji zz ,,
If is not positive semi-definite, it has a Negative eigenvalue. Let be the Eigenvector. Then Is a separating H.P.
Zv
0)( ZvvZvv TT
Any constraint violated is a S.H
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SDP
The SDP Problem :
n
ii
MZZ
kidZDts
ZCMax
,0
)..1(.
Finally, the SDP for Max-Cut has a well defined Dual problem. Which is another SDP program with the same objective Value. Intersecting the Primal and Dual program Creates a convex set, which is not emptyIf the program is feasible, and containsonly optimal points.
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Some examples :
2v
1v3v
1v
,..., 32 vv
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Some examples :
321 ,, vvv
654 ,, vvv
2v
1v
3v
5v
4v
6v
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Some examples :
1v
2v
3v
2v
1v
3v
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Some examples :
2v
1v
3v
1 1
1000
1v
2v
3v
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SDP
Use SDP to -approximate MAX-2SAT
The input is a 2-CNF formula, over variables .nxx ,...,1
Need to find an assignment so that the weight of the satisfied clauses is maximal.
A weight to each clause, mjCw j ..1),(
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SDP
Use SDP to -approximate MAX-2SAT
Assign a {-1,1} variables, nyy ,...,1
Also add a special {-1,1} variable , which will determine the mapping between {-1,1} to {True/False}
0y
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SDP
Use SDP to -approximate MAX-2SATGiven any boolean formula C, we want v(C) to be 1 if the formula is true,0 otherwise.
For example if thenixC 2
1)( 0 iyyCv
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SDP
Use SDP to -approximate MAX-2SAT
4
1
4
1
4
1
)3(4
12
1
2
11)(1)(
00
2000
00
jiji
jiji
jijiji
yyyyyy
yyyyyyy
yyyyxxvxxv
Another example :
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SDP
Use SDP to -approximate MAX-2SAT
This way we can change the 2-CNF to a QP in the form :
}1,1{.
)1()1(max ,,
i
jijiji
jiji
yts
yybyya
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SDP
Use SDP to -approximate MAX-2SAT
Relax the program to
1.
),1(),1(max ,,
i
jijiji
jiji
vts
vvbvva
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SDP
Use SDP to -approximate MAX-2SAT
The expected weight E[V] :
jijiji
jijiji
rvsignrvsignprb
rvsignrvsignpraVE
)),(),((2
)),(),((2][
,
,
And the same analysis will work here to show that this algorithm is an -approximate.
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Semi-Definite Algorithm for Max-CUT
Ran BerenfeldMay 10,2005