This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ===== 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: <code perl> > $K4=new GraphAdjacency(4); > for (my $i=0; $i<4; ++$i) { > for (my $j=$i+1; $j<4; ++$j) { > $K4->edge($i,$j); > } > } </code> (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: <code perl> > sub node_edge_incidences { > my $g=shift; > my $A=new Matrix<Int>($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; > } </code> Now we can construct the node-edge-incidence matrix of our graph ''%%K4%%'': <code perl> > $A=node_edge_incidences($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 </code> With this we can now construct the constraint matrix consisting of an upper part for the nonnegativity constraints $x_e \ge 0$ ... <code perl> > $I=new Matrix<Int>([[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<Int>(new Vector<Int>([0,0,0,0,0,0]) | $I); </code> ... and a lower part for the constraints $\Sigma_e x_e \le 1$ for each vertex $v \in V$, where the sum is over all edges e containing v: <code perl> > $Block2=new Matrix<Int>(new Vector<Int>([1,1,1,1]) | -$A); </code> Now we can put both parts together and define the polytope: <code perl> > $Ineqs=new Matrix<Rational>($Block1 / $Block2); > $P=new Polytope<Rational>(INEQUALITIES=>$Ineqs); </code> The matching polytope of ''%%K4%%'' is the integer hull of ''%%P%%'': <code perl> > $P_I=new Polytope<Rational>(POINTS=>$P->LATTICE_POINTS); </code> We can analyse some elementary properties of ''%%P_I%%'' ... <code perl> > 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 0 0 0 0 1 0 1 -1 -1 -1 0 0 0 1 -1 -1 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 -1 0 -1 0 -1 0 1 -1 0 0 -1 -1 0 1 0 0 -1 0 -1 -1 1 0 0 0 -1 -1 -1 0 1 0 0 0 0 0 1 0 -1 -1 0 0 -1 1 0 -1 0 -1 0 -1 > print $P_I->N_FACETS; 14 </code> ... and compare them with the according properties of the defining polytope ''%%P%%'': <code perl> > print $P->VERTICES; 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1/2 1/2 1/2 1 0 0 0 0 0 1 1 0 1/2 1/2 0 0 1/2 1 0 0 1 0 0 0 1 1/2 1/2 0 1/2 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1/2 0 1/2 0 1/2 0 > print $P->VOLUME; 1/72 > print $P_I->VOLUME; 1/90 </code> Next we analyse the combinatorics of ''%%P_I%%'': <html><img src=/lib/exe/fetch.php?media=tutorials/release/4.6/matching_polytopes/gale.png width=300 alt="Gale diagram" class="center-block"></html> <code perl> > 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 8 8 8 6 6 6 6 8 6 6 > $facet0=facet($P_I,0); > print $facet0->AMBIENT_DIM, " ", $facet0->DIM; 6 5 > print rows_labeled($facet0->VERTICES_IN_FACETS); 0:0 2 3 4 5 7 1:3 4 5 6 7 2:2 4 5 6 7 3:0 1 3 5 6 7 4:0 1 2 5 6 7 5:0 1 2 3 4 7 6:1 3 4 6 7 7:1 2 4 6 7 8:0 1 2 3 4 5 6 > $facet0->GALE; </code> The Gale diagram of ''%%facet0%%'' is depicted on the right. user_guide/tutorials/release/4.6/matching_polytopes.txt Last modified: 2022/01/14 11:15by 127.0.0.1