setOnesForest -- sets to 1 variables in a symbolic slack matrix which corresponding to edges of a spanning forest

Synopsis

Usage:

(Y, F) = setOnesForest X
Inputs:
- X, a matrix, a symbolic slack matrix
Outputs:
- Y, a matrix, the matrix X with the variables corresponding to the edges of a spanning forest set to 1
- F, a graph, spanning forest of the bipartite (non-incidence) graph whose vertices are the vertices and facets of a polytope P, and whose edges are the vertex-facet pairs of P such that the vertex is not on the facet

Description

Since row and column scaling operations preserve the projective equivalence class of a slack matrix, it is sometimes advantageous to choose a certain equivalence class representative by setting some variables in a slack matrix to 1. This reduces the computational time of calculating the slack ideal, for example.

This function automates the selection of a spanning forest in the bipartite graph corresponding to S and sets the associated variables to 1.

i1 : V = {{0, 0}, {0, 1}, {1, 1}, {1, 0}};

i2 : X = symbolicSlackMatrix V

Order of vertices is 
{{0, 0}, {1, 0}, {0, 1}, {1, 1}}

o2 = | 0   x_0 0   x_1 |
     | x_2 0   0   x_3 |
     | 0   x_4 x_5 0   |
     | x_6 0   x_7 0   |

                        4                 4
o2 : Matrix (QQ[x ..x ])  <-- (QQ[x ..x ])
                 0   7             0   7

i3 : (Y, F) = setOnesForest X

o3 = (| 0 1   0 1 |, Graph{"edges" => {{y , y }, {y , y }, {y , y }, {y ,
      | 1 0   0 1 |                      1   4     3   4     0   5     2 
      | 0 x_4 1 0 |        "ring" => QQ[y ..y ]
      | 1 0   1 0 |                      0   7
                           "vertices" => {y , y , y , y , y , y , y , y }
                                           0   1   2   3   4   5   6   7
     ------------------------------------------------------------------------
     y }, {y , y }, {y , y }, {y , y }}})
      6     3   6     0   7     1   7

o3 : Sequence

Ways to use setOnesForest :

setOnesForest(Matrix)

For the programmer

The object setOnesForest is a method function.

setOnesForest -- sets to 1 variables in a symbolic slack matrix which corresponding to edges of a spanning forest

Synopsis

Description

See also

Ways to use setOnesForest :

For the programmer