Note, this software was originally written by
Ambros Marzetta
when he was a Ph.D. student at ETHZ Zurich.
Since Ambros is no longer involved in this area of research,
I have taken over maintaining this code.
David Bremner
[back to the general information on primal-dual vertex enumeration]
Compiling pd
Download the package [gzipped tarfile of
source code]. It contains a makefile. You may want to use another compiler by changing the line
CC = gcc
Or you may want to use other compiler options by changing the line
CFLAGS = -O -DNDEBUG -DOMIT_TIMES
If you remove -DOMIT_TIMES, the program measures the execution time for the different phases of the algorithm.
Then, simply type
make
and you get an executable named pd.
Running pd
pd reads the input data from stdin. Start the program by typing
% pd < pdtest/cross2.ine
Input format
pd accepts files in .ine-format like lrs, prs and
cdd. The
input must consist of rational numbers (no floating point
numbers). There are some restrictions on the input data:
- V-representation: The polyhedron must be bounded (no rays) and full-dimensional.
- H-representation: The origin must be in the interior of the polyhedron (in
the interior, not on the border!), and the polyhedron must have at least one vertex.
There may be a digits command before the begin line:
- digits n: Use n decimal digits for the numerator and denominator of all rational numbers. Default is 180.
There may be options after the end of the input matrix:
- incidence: After every facet found, print a list of its incident vertices.
- #incidence: After every facet found, print the number of incident vertices.
- notranslation: For V-representations: The origin is in the
interior of the polytope. For H-representations: The input polyhedron
is bounded. In both cases, the option instructs the program not to
translate the input. This can make the computation much faster.
- volume: Compute the volume of the polytope. This option is only
valid for H-representations and if the polyhedron is bounded.
Example 1: Vertex enumeration of a cube
Input
Cube of dimension 3 (the center is at the origin)
H-representation
digits 20
begin
6 4 integer
1 1 0 0
1 0 1 0
1 0 0 1
1 -1 0 0
1 0 -1 0
1 0 0 -1
end
notranslation
The output should contain (along with other comment lines beginning with a "*")
dec digits: 20
*Input File: Cube
V-representation
begin
***** 4 rational
1 -1 1 1
1 -1 -1 1
1 -1 1 -1
1 1 1 1
1 1 -1 -1
1 -1 -1 -1
1 1 1 -1
1 1 -1 1
end
*Number of vertices+rays: 8
Example 2: Facet enumeration of a pyramid
Input
Pyramid (the last vertex is the top of the pyramid)
V-representation
begin
5 4 integer
1 10 10 0
1 10 12 0
1 12 10 0
1 12 12 0
1 11 11 5
end
incidence
The output should contain (along with other comment lines beginning with
a "*")
*Input File: Pyramid
H-representation
begin
***** 4 rational
60 -5 0 -1: 5 4 3|3
60 0 -5 -1: 5 4 2|3
-50 5 0 -1: 5 2 1|3
0 0 0 1: 4 3 2 1|4
-50 0 5 -1: 5 3 1|3
end
*Number of facets: 5
The numbers between the ":" and the "|" give the vertices incident to the facet, and the number after the "|" gives the number of incident vertices.
Error messages
When pd is run, the following error messages may be generated:
- Syntax error in the input file.
- Internal error. Please send me a bug report.
- Not enough memory. Change (decrease) the digits line in your input file.
- Overflow. Change (increase) the digits line in your input file.
- Digits is too large. Increase MAX_DIGITS in prs_mp.h and recompile the
program.
- Unbounded / origin not in the interior. Remove the "notranslation" or
the "volume" option from the input file, and check the restrictions on
the input data.
- Not full-dimensional.
Release notes
pd prints its revision number in the first or second line of the output. Currently, this is revision 1.7.
Bug reports
Send bug reports to bremner@unb.ca.
97-11-20 / PD Home