maxcut.gms : Goemans/Williamson Randomized Approximation Algorithm for MaxCut

**Description**

Let G(N, E) denote a graph. A cut is a partition of the vertices N into two sets S and T. Any edge (u,v) in E with u in S and v in T is said to be crossing the cut and is a cut edge. The size of the cut is defined to be sum of weights of the edges crossing the cut. This model presents a simple MIP formulation of the problem that is seeded with a solution from the Goemans/Williamson randomized approximation algorithm based on a semidefinite programming relaxation. The MaxCut instance tg20_7777 is available from the Biq Mac Library and comes from applications in statistical physics.

**References**

- Goemans M.X., and Williamson, D P, Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. Journal of the ACM 42 (1995), 1115-1145.
- Wiegele A., Biq Mac Library - Binary quadratic and Max cut Library.

**Large Model of Type :** MIP

**Category :** GAMS Model library

**Main file :** maxcut.gms **includes :** runcsdp.inc tg207777.inc

```
$title Goemans/Williamson Randomized Approximation Algorithm for MaxCut (MAXCUT,SEQ=338)
$onText
Let G(N, E) denote a graph. A cut is a partition of the vertices N
into two sets S and T. Any edge (u,v) in E with u in S and v in T is
said to be crossing the cut and is a cut edge. The size of the cut is
defined to be sum of weights of the edges crossing the cut.
This model presents a simple MIP formulation of the problem that is
seeded with a solution from the Goemans/Williamson randomized
approximation algorithm based on a semidefinite programming
relaxation.
The MaxCut instance tg20_7777 is available from the Biq Mac Library
and comes from applications in statistical physics.
Wiegele A., Biq Mac Library - Binary Quadratic and Max Cut Library.
http://biqmac.uni-klu.ac.at/biqmaclib.html
Goemans M.X., and Williamson, D.P., Improved Approximation Algorithms
for Maximum Cut and Satisfiability Problems Using Semidefinite
Programming. Journal of the ACM 42 (1995), 1115-1145.
http://www-math.mit.edu/~goemans/PAPERS/maxcut-jacm.pdf
Keywords: mixed integer linear programming, approximation algorithms,
convex optimization, randomized algorithms, maximum cut problem,
mathematics
$offText
Set n 'nodes';
Alias (n,i,j);
Parameter w(i,j) 'edge weights';
Set e(i,j) 'edges';
$if not set instance $set instance tg207777.inc
* Simple AWK script to convert MAXCUT format to GAMS format
$onEcho > maxcut.awk
NR==1 { print "set n /1*" $1 "/";
print "parameter w(n,n) /\n$ondelim" }
NR>1 { print $0 }
END { print "\n$offdelim\n/;" }
$offEcho
$call awk -f maxcut.awk %instance% > %instance%.gms
$offListing
$include %instance%.gms
$onListing
$eval maxn card(n)
* We want all edges to be i-j with i<j;
e(i,j) = ord(i) < ord(j);
w(e(i,j)) = w(i,j) + w(j,i);
w(i,j)$(not e(i,j)) = 0;
option e < w;
* Simple MIP model
Variable
x(n) 'decides on what side of the cut'
cut(i,j) 'edge is in the cut'
z 'objective';
Binary Variable x;
Equation obj, xor1(i,j), xor2(i,j), xor3(i,j), xor4(i,j);
obj.. z =e= sum(e, w(e)*cut(e));
xor1(e(i,j)).. cut(e) =l= x(i) + x(j);
xor2(e(i,j)).. cut(e) =l= 2 - x(i) - x(j);
xor3(e(i,j)).. cut(e) =g= x(i) - x(j);
xor4(e(i,j)).. cut(e) =g= x(j) - x(i);
Model maxcut / all /;
$onText
Set up the SDP
max W*Y s.t. Y_ii = 1, Y > 0
We need to pass on the dual to csdp
min x1 + x2 + ... + xn s.t. X = F1*x1 + F2*x2 + ... + Fn*xn - W, X > 0
with F_i = 1 for F_ii and 0 otherwise
$offText
Parameter
c(n) 'cost coefficients'
F(n,i,j) 'constraint matrix'
F0(i,j) 'constant term'
Y(i,j) 'dual solution'
L(i,j) 'Cholesky factor of dual solution Y';
c(n) = 1;
F(n,n,n) = 1;
F0(i,j) = -w(i,j);
execute_unload 'csdpin.gdx' n = m, n, c, F, F0;
execute 'gams runcsdp.inc lo=%gams.lo% --strict=1 && cholesky csdpout.gdx n Y cholesky.gdx L ';
abort$errorLevel 'Problems running RunCSDP. Check listing file for details.';
execute_load 'cholesky.gdx' L;
execute_load 'csdpout.gdx' Y;
Scalar SDPRelaxation;
SDPRelaxation = 0.5*sum(e, w(e)*(1 - Y(e)));
display SDPRelaxation;
* Check if Cholesky factorization is correct
Parameter Y_, Ydiff;
Y_(i,j) = sum(n, L(i,n)*L(j,n));
Ydiff(i,j) = round(Y(i,j) - Y_(i,j),1e-6);
option Ydiff:8:0:1;
abort$card(Ydiff) Ydiff;
* Now do the random hyperplane r
Parameter r(n);
Set S(n), T(n), bestS(n);
Scalar
wS 'weight of cut S'
maxwS 'best weight' / -inf /
cnt;
for(cnt = 1 to 10,
r(n) = uniform(-1,1);
S(n) = sum(i, L(n,i)*r(i)) < 0;
T(n) = yes;
T(S) = no;
wS = sum(e(S,T), w(S,T) + w(T,S));
if(wS > maxwS, maxwS = wS; bestS(n) = S(n););
);
display maxwS;
* use computed feasible solution as starting point for MIP solve
x.l(bestS) = 1;
cut.l(e(i,j)) = x.l(i) xor x.l(j);
* SCIP does this by default, for other solvers we need to enable it
$if %gams.mip% == cplex $echo mipStart 1 > cplex.opt
$if %gams.mip% == cbc $echo mipStart 1 > cbc.opt
$if %gams.mip% == gurobi $echo mipStart 1 > gurobi.opt
$if %gams.mip% == xpress $echo loadmipsol 1 > xpress.opt
maxcut.optFile = 1;
solve maxcut max z using mip;
```