user_guide:tutorials:latest:apps_matroid

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user_guide:tutorials:latest:apps_matroid [2020/01/22 09:02] (current)
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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 ''​%%%%set_custom $default_application="​matroid";​%%%%''​
 +
 +===== 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;​
 +3 4 3
 +</​code>​
 +{{ :​tutorial:​matroid//​lattice//​of//​flats//​example.png?​nolink&​200|}} 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}
 +{1 3}
 +{0 3}
 +{0 1}
 +</​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}
 +{0 2}
 +{1 2}
 +{2 3}
 +> $M->​LATTICE_OF_FLATS->​VISUAL;​
 +</​code>​
 +===== Matroid Polytopes =====
 +
 +You can construct a polytope from the bases of a matroid as the convex hull of the characteristic vectors of the bases. This is the //matroid polytope// of that matroid, sometimes also called the //matroid bases polytope//. The matroid polytope of the matroid ''​$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: 2020/01/22 09:02
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