semi-definite algorithm for max-cut ran berenfeld may 10,2005
DESCRIPTION
Semi-Definite Algorithm for Max-CUT Ran Berenfeld May 10,2005. Agenda : The Max-Cut problem. Goemans-Williamson algorithm. Semi-Definite programming. Other applications. The Max-Cut Problem : Let be a complete, undirected graph, With edge weights . - PowerPoint PPT PresentationTRANSCRIPT
Semi-Definite Algorithm for Max-CUT
Ran BerenfeldMay 10,2005
Agenda :
•The Max-Cut problem.•Goemans-Williamson algorithm.•Semi-Definite programming.•Other applications.
The Max-Cut Problem :
Let be a complete, undirected graph,
With edge weights .
Find a cut that maximizes
EVG ,
QEW :
S
SwSv
wve
ew
,,
)(
•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
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
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
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
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
VPSDP
we’ll show later that VP is equal to SDP.
SDP
we’ll also show later how SDP is polynomial time solvable to any accuracy degree.But first lets analyze the approximation ratio.
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
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 :
),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
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 ][
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
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
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 :
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.
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…
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 ,
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
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.
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
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.
Some examples :
2v
1v3v
1v
,..., 32 vv
Some examples :
321 ,, vvv
654 ,, vvv
2v
1v
3v
5v
4v
6v
Some examples :
1v
2v
3v
2v
1v
3v
Some examples :
2v
1v
3v
1 1
1000
1v
2v
3v
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),(
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
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
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 :
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
SDP
Use SDP to -approximate MAX-2SAT
Relax the program to
1.
),1(),1(max ,,
i
jijiji
jiji
vts
vvbvva
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.
Semi-Definite Algorithm for Max-CUT
Ran BerenfeldMay 10,2005