forum.polymake.org

Posted: **11 Feb 2015, 15:33**

Hello again,

I am using function gc_closure on rational polytope and on following polytope (2D with 4 vertices)

$q=new Polytope<Rational>(VERTICES=>[[1,1,0],[1,2,15/10],[1,24/10,15/10],[1,3,0]],LINEALITY_SPACE=>[[]]);
$qq=gc_closure($q);
print $qq->VERTICES;

it returns correct gomory chvatal closure.

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1 1 0
1 3 0
1 2 1

When called on slightly modified polytope (3rd vertex is slightly moved)

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$q=new Polytope<Rational>(VERTICES=>[[1,1,0],[1,2,15/10],[1,25/10,15/10],[1,3,0]],LINEALITY_SPACE=>[[]]);
$qq=gc_closure($q);
print $qq->VERTICES;

it returns following closure which should be identical to the previous one.

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Am I using something in incorrect way or is it a bug?

Thanks for info.
Peter

Edit: using polymake version 2.13

Posted: **11 Feb 2015, 17:20**

Hi there,

without investigating your examples in great detail, I think there is a misunderstanding here.

The function \(\texttt{gc_closure}\) does not compute the integer hull. It just does one iteration step. Doing the first gomory chvatal closure means: take all inequalities induced by the hilbert basis of the normal cones of each vertex and floor the right hand side (after you made sure the coefficients on the left side are integral).

If we repeat this process we will obtain the integer hull. Indeed if we call

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$q=new Polytope<Rational>(VERTICES=>[[1,1,0],[1,2,15/10],[1,25/10,15/10],[1,3,0]],LINEALITY_SPACE=>[[]]);
$qq=gc_closure(gc_closure($q));
print $qq->VERTICES;
1 3 0
1 1 0
1 2 1

we get the same vertices as for your first example.

Since your second polytope has different inequalities and different normal cones I don't see why gc_closure should produce the same result. For me your output looks plausible.

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