===== Matching Polytopes ===== In this tutorial we will use ''polymake'' to construct and analyse matching polytopes. First we construct a graph, the complete graph on four nodes:


> $K4=new props::Graph(4);
> for (my $i=0; $i<4; ++$i) {
>   for (my $j=$i+1; $j<4; ++$j) {
>     $K4->edge($i,$j);
>   }
> }

(See also the [[apps_graph|Tutorial on Graphs]] for more on the construction of graphs.) Next we like to have the node-edge-incidence matrix of our graph. Since the latest release of ''polymake'' does not yet support this, we have to write the function ourselves:


> sub node_edge_incidences {
>     my $g=shift;
>    my $A=new Matrix($g->nodes, $g->edges);
>     my $k=0;
>    for (my $i=0; $i<$g->nodes-1; ++$i) {
>        foreach (@{$g->adjacent_nodes($i)}) {
>            if ($_>$i) {
>                $A->[$i]->[$k]=1;
>                $A->[$_]->[$k]=1;
>                ++$k;
>            }
>        }
>    }
>    return $A;
> }

Now we can construct the node-edge-incidence matrix of our graph ''K4'':


> $A=node_edge_incidences($K4);
> print $A;
1 1 1 0 0 0
1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1

With this we can now construct the constraint matrix consisting of an upper part for the nonnegativity constraints x_e≥0 ...


> $I=new Matrix([[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]);
> $Block1=new Matrix(new Vector([0,0,0,0,0,0]) | $I);

... and a lower part for the constraints Σ_e x_e≤1 for each vertex v∈V, where the sum is over all edges e containing v:


> $Block2=new Matrix(new Vector([1,1,1,1]) | -$A);

Now we can put both parts together and define the polytope:


> $Ineqs=new Matrix($Block1 / $Block2);
> $P=new Polytope(INEQUALITIES=>$Ineqs);

The matching polytope of ''K4'' is the integer hull of ''P'':


> $P_I=new Polytope(POINTS=>$P->LATTICE_POINTS);

We can analyse some elementary properties of ''P_I'' ...


> print $P_I->POINTS;
1 0 0 0 0 0 0
1 0 0 0 0 0 1
1 0 0 0 0 1 0
1 0 0 0 1 0 0
1 0 0 1 0 0 0
1 0 0 1 1 0 0
1 0 1 0 0 0 0
1 0 1 0 0 1 0
1 1 0 0 0 0 0
1 1 0 0 0 0 1
> print $P_I->FACETS;
0 0 0 0 0 0 1
0 1 0 0 0 0 0
1 0 0 0 -1 -1 -1
1 -1 0 0 -1 -1 0
1 0 -1 0 -1 0 -1
1 -1 -1 0 -1 0 0
1 0 0 -1 0 -1 -1
1 -1 0 -1 0 -1 0
1 0 -1 -1 0 0 -1
1 -1 -1 -1 0 0 0
0 0 0 0 0 1 0
0 0 1 0 0 0 0
0 0 0 0 1 0 0
0 0 0 1 0 0 0
> print $P_I->N_FACETS;
14

... and compare them with the according properties of the defining polytope ''P'':


> print $P->VERTICES;
1 0 0 0 1 0 0
1 0 1 0 0 0 0
1 1/2 1/2 0 1/2 0 0
1 0 0 0 0 0 0
1 1 0 0 0 0 0
1 1/2 0 1/2 0 1/2 0
1 0 1/2 1/2 0 0 1/2
1 0 0 0 1/2 1/2 1/2
1 0 0 0 0 1 0
1 0 0 1 0 0 0
1 0 0 0 0 0 1
1 1 0 0 0 0 1
1 0 1 0 0 1 0
1 0 0 1 1 0 0
> print $P->VOLUME;
1/72
> print $P_I->VOLUME;
1/90

Next we analyse the combinatorics of ''P_I'': {{ :tutorial:ilp:gale.png?300|The Gale diagram of ''facet0''}}


> print $P_I->AMBIENT_DIM, " ", $P_I->DIM;
6 6
> print $P_I->F_VECTOR;
10 39 78 86 51 14
> print $P_I->FACET_SIZES;
8 8 6 6 6 6 6 6 6 6 8 8 8 8
> $facet0=facet($P_I,0);
> print $facet0->AMBIENT_DIM, " ", $facet0->DIM;
6 5
> print rows_labeled($facet0->VERTICES_IN_FACETS);
0:0 1 2 3 4 5 6
1:1 2 4 6 7
2:2 4 5 6 7
3:1 3 4 6 7
4:3 4 5 6 7
5:0 2 3 4 5 7
6:0 1 2 3 4 7
7:0 1 3 5 6 7
8:0 1 2 5 6 7
> $facet0->GALE;

The Gale diagram of ''facet0'' is depicted on the right.