documentation:latest:polytope:propagatedpolytope

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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

# BigObject PropagatedPolytope<Scalar>

from application polytope

Polytope propagation means to define a polytope inductively by assigning vectors to arcs of a directed graph. At each node of such a graph a polytope arises as the joint convex hull of the polytopes at the translated sources of the inward pointing arcs. For details see

Joswig: Polytope Propagation on Graphs.
Chapter 6 in Pachter/Sturmfels: Algebraic Statistics for Computational Biology, Cambridge 2005.
Type Parameters:

Scalar: numeric data type used for the coordinates, must be an ordered field. Default is Rational.

derived from:
Polytope
Example:

We define an acyclic digraph on 8 nodes with a unique source and a unique sink. Each arc is equipped with a vector in Q^2.

 > $Gamma = new GraphAdjacency<Directed>(8); >$Vectors = new EdgeMap<Directed, Vector>($Gamma); >$Vectors->edge(0,1) = [1,0]; $Vectors->edge(0,2) = [0,0]; >$Vectors->edge(1,3) = [1,0]; $Vectors->edge(1,4) = [0,1]; >$Vectors->edge(2,3) = [0,0]; $Vectors->edge(2,4) = [0,2]; >$Vectors->edge(3,5) = [1,0]; $Vectors->edge(3,6) = [0,1]; >$Vectors->edge(4,5) = [0,0]; $Vectors->edge(4,6) = [0,2]; >$Vectors->edge(5,7) = [0,0];
> $Vectors->edge(6,7) = [0,0]; Then we can define the propagated polytope implicitly:  >$P = new PropagatedPolytope(CONE_AMBIENT_DIM=>3, "SUM_PRODUCT_GRAPH.ADJACENCY"=>$Gamma, "SUM_PRODUCT_GRAPH.TRANSLATIONS"=>$Vectors);
> print \$P->VERTICES;
1 3 0
1 1 0
1 0 1
1 1 3
1 0 4

Note that it is necessary to specify the dimension of the (homogeneous) ambient space by setting CONE_AMBIENT_DIM.

SUM_PRODUCT_GRAPH
Type:
Graph<Directed>
Properties of SUM_PRODUCT_GRAPH:
TRANSLATIONS

The translation vectors of the arcs.

Type:
EdgeMap<Directed,Vector<Scalar>>

• documentation/latest/polytope/propagatedpolytope.txt