GAP (CddInterface) - Chapter 1: Introduction

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1 Introduction
1.1 Why CddInterface
1.2 H-representation and V-representation of polyhedra

### 1 Introduction

#### 1.1 Why CddInterface

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:

4-3x_1+6x_2-5x_4 = 0, 1+2x_1-2x_2-7x_3 \geq 0, -3x_2+5x_4 = 0;

is the following:

    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} ).

Then the Polyhedra V-format is for P is:

    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.

For example the V-representation of the polyhedron defined as

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

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