/usr/share/polymake/demo/matching_polytopes.ipynb is in polymake-common 3.2r2-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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"cells": [
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Matching Polytopes\n",
"\n",
"In this tutorial we will use `polymake` to construct and analyse matching polytopes.\n",
"\n",
"First we construct a graph, the complete graph on four nodes:\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
">\n"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"$K4=new props::Graph(4);"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"for (my $i=0; $i<4; ++$i) {\n",
" for (my $j=$i+1; $j<4; ++$j) {\n",
" $K4->edge($i,$j);\n",
" }\n",
"}"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"(See also the [Tutorial on Graphs](apps_graph) for more on the construction of graphs.)\n",
"\n",
"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:\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"sub node_edge_incidences {\n",
" my $g=shift;\n",
" my $A=new Matrix<Int>($g->nodes, $g->edges);\n",
" my $k=0;\n",
" for (my $i=0; $i<$g->nodes-1; ++$i) {\n",
" foreach (@{$g->adjacent_nodes($i)}) {\n",
" if ($_>$i) {\n",
" $A->[$i]->[$k]=1;\n",
" $A->[$_]->[$k]=1;\n",
" ++$k;\n",
" }\n",
" }\n",
" }\n",
" return $A;\n",
"}"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"Now we can construct the node-edge-incidence matrix of our graph `K4`:\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 1 1 0 0 0\n",
"1 0 0 1 1 0\n",
"0 1 0 1 0 1\n",
"0 0 1 0 1 1\n",
"\n"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"$A=node_edge_incidences($K4);\n",
"print $A;"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"With this we can now construct the constraint matrix consisting of an upper part for the nonnegativity constraints x<sub>e</sub><html>≥</html>0 ...\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"$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]]);\n",
"$Block1=new Matrix<Int>(new Vector<Int>([0,0,0,0,0,0]) | $I);"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"... and a lower part for the constraints <html>Σ</html><sub>e</sub> x<sub>e</sub><html>≤</html>1 for each vertex v<html>∈</html>V, where the sum is over all edges e containing v:\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"$Block2=new Matrix<Int>(new Vector<Int>([1,1,1,1]) | -$A);"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"Now we can put both parts together and define the polytope:\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"$Ineqs=new Matrix<Rational>($Block1 / $Block2);\n",
"$P=new Polytope<Rational>(INEQUALITIES=>$Ineqs);"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"The matching polytope of `K4` is the integer hull of `P`:\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"$P_I=new Polytope<Rational>(POINTS=>$P->LATTICE_POINTS);"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"We can analyse some elementary properties of `P_I` ...\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 0 0 0 0 0 0\n",
"1 0 0 0 0 0 1\n",
"1 0 0 0 0 1 0\n",
"1 0 0 0 1 0 0\n",
"1 0 0 1 0 0 0\n",
"1 0 0 1 1 0 0\n",
"1 0 1 0 0 0 0\n",
"1 0 1 0 0 1 0\n",
"1 1 0 0 0 0 0\n",
"1 1 0 0 0 0 1\n",
" \n"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P_I->POINTS;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0 0 0 0 0 0 1\n",
"0 1 0 0 0 0 0\n",
"1 0 0 0 -1 -1 -1\n",
"1 -1 0 0 -1 -1 0\n",
"1 0 -1 0 -1 0 -1\n",
"1 -1 -1 0 -1 0 0\n",
"1 0 0 -1 0 -1 -1\n",
"1 -1 0 -1 0 -1 0\n",
"1 0 -1 -1 0 0 -1\n",
"1 -1 -1 -1 0 0 0\n",
"0 0 0 0 0 1 0\n",
"0 0 1 0 0 0 0\n",
"0 0 0 0 1 0 0\n",
"0 0 0 1 0 0 0\n",
" \n"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P_I->FACETS;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"14\n",
"\n"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P_I->N_FACETS;"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"... and compare them with the according properties of the defining polytope `P`:\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 0 0 0 1 0 0\n",
"1 0 1 0 0 0 0\n",
"1 1/2 1/2 0 1/2 0 0\n",
"1 0 0 0 0 0 0\n",
"1 1 0 0 0 0 0\n",
"1 1/2 0 1/2 0 1/2 0\n",
"1 0 1/2 1/2 0 0 1/2\n",
"1 0 0 0 1/2 1/2 1/2\n",
"1 0 0 0 0 1 0\n",
"1 0 0 1 0 0 0\n",
"1 0 0 0 0 0 1\n",
"1 1 0 0 0 0 1\n",
"1 0 1 0 0 1 0\n",
"1 0 0 1 1 0 0\n",
" \n"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P->VERTICES;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1/72\n",
" \n"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P->VOLUME;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1/90\n",
"\n"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P_I->VOLUME;"
]
},
{
"attachments": {
"gale.png": {
"image/png": [
"Tm90IEZvdW5k"
]
}
},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"Next we analyse the combinatorics of `P_I`:\n",
"\n",
"\n",
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6 6\n",
" \n"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P_I->AMBIENT_DIM, \" \", $P_I->DIM;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10 39 78 86 51 14\n",
" \n"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P_I->F_VECTOR;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8 8 6 6 6 6 6 6 6 6 8 8 8 8\n",
" \n"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $P_I->FACET_SIZES;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"$facet0=facet($P_I,0);"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
" \n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"6 5\n",
" \n"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print $facet0->AMBIENT_DIM, \" \", $facet0->DIM;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0:0 1 2 3 4 5 6\n",
"1:1 2 4 6 7\n",
"2:2 4 5 6 7\n",
"3:1 3 4 6 7\n",
"4:3 4 5 6 7\n",
"5:0 2 3 4 5 7\n",
"6:0 1 2 3 4 7\n",
"7:0 1 3 5 6 7\n",
"8:0 1 2 5 6 7\n",
" \n"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"print rows_labeled($facet0->VERTICES_IN_FACETS);"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"$facet0->GALE;"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"The Gale diagram of `facet0` is depicted on the right.\n"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "polymake",
"language": "polymake",
"name": "polymake"
},
"language_info": {
"codemirror_mode": "perl",
"file_extension": ".pm",
"mimetype": "text/x-polymake",
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