user_guide:tutorials:latest:apps_matroid

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—user_guide:tutorials:latest:apps_matroid [2023/11/06 10:57] (current) – created - external edit 127.0.0.1
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 +====== Short Introduction to application matroid ======
 +
 +This tutorial is meant to show the main features for handling matroids available. To make ''%%matroid%%'' your current application start ''%%polymake%%'' with the option ''%%-A matroid%%'' or use the context switch
 +
 +<code perl>
 +> application "matroid";
 +</code>
 +from within the ''%%polymake%%'' shell. A permanent setting can be stored with
 +
 +<code perl>
 +set_custom $default_application="matroid";
 +</code>
 +===== Constructing a Simple Matroid and Playing Around =====
 +
 +This is how to produce a matroid from a vector configuration over the rationals. The matroid is defined by the linear dependence among subsets of these vectors.
 +
 +<code perl>
 +> $M=new Matroid(VECTORS=>[[1,0,0],[1,0,1],[1,1,0],[1,0,2]]);
 +</code>
 +If ''%%matroid%%'' is not your default application you have to qualify ''%%Matroid%%'' as in:
 +
 +<code perl>
 +> $M=new matroid::Matroid(VECTORS=>[[1,0,0],[1,0,1],[1,1,0],[1,0,2]]);
 +</code>
 +Output of basic statistics.
 +
 +<code perl>
 +> print $M->N_BASES, " ", $M->N_ELEMENTS, " ", $M->RANK;
 +> svg($M->LATTICE_OF_FLATS->VISUAL);
 +3 4 3
 +</code>
 +{{ tutorials:release:4.11:apps_matroid:output_0.svg }}
 +The ''%%VECTORS%%'' are numbered consecutively, starting from zero. The bases are encoded as sets of these ordinal numbers.
 +
 +<code perl>
 +> print $M->BASES;
 +{0 1 2}
 +{0 2 3}
 +{1 2 3}
 +</code>
 +Similarly you can compute the circuits and cocircuits.
 +
 +<code perl>
 +> print $M->CIRCUITS;
 +{0 1 3}
 +> print $M->COCIRCUITS;
 +{2}
 +{0 1}
 +{0 3}
 +{1 3}
 +</code>
 +You can also compute other properties, like
 +
 +<code perl>
 +> print $M->PAVING?"1":"0", " ",
 +> $M->BINARY?"1":"0", " ",
 +> $M->SERIES_PARALLEL?"1":"0", " ",
 +> $M->CONNECTED?"1":"0";
 +1 1 0 0
 +> print $M->CONNECTED_COMPONENTS;
 +{0 1 3}
 +{2}
 +> print $M->TUTTE_POLYNOMIAL;
 +x_0^3 + x_0^2 + x_0*x_1
 +</code>
 +Even the lattice of flats could be computed and visualised.
 +
 +<code perl>
 +> $lattice=$M->LATTICE_OF_FLATS;
 +> foreach (@{$lattice->nodes_of_rank(2)}){print $lattice->FACES->[$_]," "};
 +{0 2} {0 1 3} {1 2} {2 3} 
 +> print $M->MATROID_HYPERPLANES;
 +{0 1 3}
 +{2 3}
 +{1 2}
 +{0 2}
 +</code>
 +===== Matroid Polytopes =====
 +
 +You can construct a polytope from the bases of a matroid as the convex hull of the characteristic vectors of the bases. This is the //matroid polytope// of that matroid, sometimes also called the //matroid bases polytope//. The matroid polytope of the matroid ''%%$M%%'' is a subobject ''%%POLYTOPE%%'' of type `polytope::Polytope<html><Rational></html>.
 +
 +<code perl>
 +> print $M->POLYTOPE->VERTICES;
 +1 1 1 1 0
 +1 1 0 1 1
 +1 0 1 1 1
 +> print $M->POLYTOPE->F_VECTOR;
 +3 3
 +</code>
 +===== Other Constructions =====
 +
 +The vertices of a polytope give rise to a matroid. Here is an example for the vertices of the three-dimensional regular cube. Notice that point coordinates in the application 'polytope' are given by homogeneous coordinates. Hence this matroid is defined by the relation of affine dependence.
 +
 +<code perl>
 +> $C=new Matroid(VECTORS=>polytope::cube(3)->VERTICES);
 +> print $C->N_BASES;
 +58
 +</code>
 +The system also allows you to construct a matroid from a graph. The bases correspond to the spanning trees then. Notice that there is more than one way to encode a graph in ''%%polymake%%''. Read the [[apps_graph|tutorial on graphs]] for details.
 +
 +<code perl>
 +> $G=matroid_from_graph(polytope::cube(3)->GRAPH);
 +> print $G->N_BASES;
 +384
 +</code>
 +It is also possible to derive a new matroid from others.
 +
 +<code perl>
 +> # The arguments are two matroids and for each matroid a basepoint. The basepoints will be identified. 
 +> $se=series_extension(uniform_matroid(2,3),0,uniform_matroid(1,3),0);
 +> print deletion($se,4)->VECTORS;
 +1 0 0
 +0 1 0
 +0 0 1
 +1 1 1
 +> $pe=parallel_extension(uniform_matroid(1,3),0,uniform_matroid(2,3),0);
 +> print dual(contraction($pe,4))->VECTORS;
 +1 1 1
 +1 0 0
 +0 1 0
 +0 0 1
 +> print projective_plane(3)->N_BASES;
 +234
 +> print fano_matroid()->N_BASES;
 +28
 +> print direct_sum(projective_plane(3),fano_matroid())->N_BASES," = 234*28";
 +6552 = 234*28
 +> print two_sum(uniform_matroid(2,4),0,uniform_matroid(2,4),0)->CIRCUITS;
 +{0 1 2}
 +{3 4 5}
 +{0 1 3 4}
 +{0 1 3 5}
 +{0 1 4 5}
 +{0 2 3 4}
 +{0 2 3 5}
 +{0 2 4 5}
 +{1 2 3 4}
 +{1 2 3 5}
 +{1 2 4 5}
 +</code>
 +Of course you can also construct your matroid from scratch by specifying, e.g., its set of bases or non-bases and then compute other properties. The following constructs the Fano matroid, which is the simplest matroid that cannot be constructed from a vector configuration (over a field with a characteristic other than two).
 +
 +<code perl>
 +> $a=new Array<Set<Int>>([0,1,5],[1,2,6],[0,2,3],[1,3,4],[2,4,5],[3,5,6],[0,4,6]);
 +> $m=new Matroid(NON_BASES=>$a,N_ELEMENTS=>7);
 +> print $m->COCIRCUITS;
 +{0 1 2 4}
 +{0 1 3 6}
 +{0 2 5 6}
 +{0 3 4 5}
 +{1 2 3 5}
 +{1 4 5 6}
 +{2 3 4 6}
 +</code>
 +Note that you have to specify N_ELEMENTS when constructing a matroid in this way because this is not implicit in BASES, etc.
 +
  
  • user_guide/tutorials/latest/apps_matroid.txt
  • Last modified: 2023/11/06 10:57
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