GAP (CddInterface) - Chapter 1: Introduction

We know that every convex polyhedron has two representations, one as the intersection of finite halfspaces and the other as Minkowski sum of the convex hull of finite points and the nonnegative hull of finite directions. These are called \(H\)-representation and \(V\)-representation, respectively. CddInterface is a gap interface to the C package cddlib which among other things can translate between these two representations.

1.2 H-representation and V-representation of polyhedra

Let us start by introducing the \(H\)-representation. Let A be m x d matrix and let b be a column m-vector. The \(H\)-representation of the polyhedron defined by the system b+Ax >= 0 of m inequalities and d variables x= (x_1,...,x_d) is as follows:

H-representation
linearity t, [i_1, i_2, ...,i_t]
begin
  m x (d+1) numbertype
  b  A
end

The linearity line is added when we want to specify that some rows of the system \(b+Ax\) are equalities. That is, \(k\in \{i_1, i_2, \dots,i_t\}\) means that the row \(k\) of the system \(b+Ax\) is specified to be equality.

For example, the \(H\)-representation of the polyhedron defined by the following system:

H-representation
linearity 2, [1, 3]
begin
3 x 5 rational
  4 -3  6  0 -5
  1  2 -2 -7  0
  0  0 -3  0  5
end

Next we define Polyhedra \(V\)-format. Let P be represented by n gerating points and s generating directions (rays) as

\[P = \mathrm{conv}(v_1 , \dots , v_n ) + \mathrm{nonneg}(r_{n+1} , \dots , r_{n+s} ).\]

V-representation
linearity t, [i_1, i_2,...,i_t]
begin
(n+s) x (d+1) numbertype
  1  v_1
  :   :
  1  v_n
  0  r_{n+1}
  :   :
  0  r_{n+s}
end

In the above format the generating points and generating rays may appear mixed in arbitrary order. Linearity for \(V\)-representation specifies a subset of generators whose coefficients are relaxed to be free. That is, \(k \in \{i_1 , i_2 , . . . , i_t \}\) specifies that the \(k\)-th generator is specified to be free. This means for each such a ray \(r_k\) , the line generated by \(r_k\) is in the polyhedron, and for each such a vertex \(v_k\) , its coefficient is no longer nonnegative but still the coefficients for all \(v_i\)’s must sum up to one.

\[P:= \mathrm{conv}( (2,3), (-2,-3), (-1,2) ) + \mathrm{nonneg}(\; (1,2) , (-1,-2), (1,1)\;)\]

V-representation
linearity 2, [ 1, 3 ]
begin
 4 x 3 rational
 1   2   3
 1  -1   2
 0   1   2
 0   1   1
end

1 Introduction

1.1 Why CddInterface

1.2 H-representation and V-representation of polyhedra